# Discovery of $\gamma$-ray pulsations of PSR J1835$-$3259B in the Globular
Cluster NGC 6652

Pengfei Zhang  Department of Astronomy, School of Physics and Astronomy, Key
Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University,
Kunming 650091, People’s Republic of China<EMAIL_ADDRESS><EMAIL_ADDRESS>Yi Xing  Key Laboratory for Research in Galaxies and
Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80
Nandan Road, Shanghai 200030, People’s Republic of China<EMAIL_ADDRESS>Zhongxiang Wang Department of Astronomy, School of Physics and Astronomy, Key
Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University,
Kunming 650091, People’s Republic of China<EMAIL_ADDRESS><EMAIL_ADDRESS>Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

Motivated by the newly discovery of a spin period 1.83 ms pulsar J1835$-$3259B
in the globular cluster (GC) NGC 6652, we analyze the $\gamma$-ray data
obtained with the Large Area Telescope (LAT) onboard Fermi Gamma-ray Space
Telescope (Fermi) for the GC and detect the pulsations of this millisecond
pulsar (MSP) at a 5.4$\sigma$ confidence level (the weighted H-test value is
$\sim 41$). From timing analysis of the data, a pulse profile that is similar
to the radio one is established. We thus consider that we have detected the
$\gamma$-ray emission of the MSP, and discuss the implications. Based on our
analysis results and different studies of the sources in the GC, the observed
$\gamma$-ray emission from the GC could mainly arise from this MSP, like the
previous two cases the GCs NGC 6624 and NGC 6626. Assuming this is the case,
the pulsar, at the GC’s 9.46 kpc distance and having a spin-down luminosity of
$\leq 4.3\times 10^{35}$ erg s-1, would have a $\gamma$-ray luminosity of
$\simeq(5.04\pm 0.44)\times 10^{34}$ erg s-1 and a $\gamma$-ray efficiency of
$\gtrsim 0.12$.

Gamma-rays(637); Globular star clusters(656); Pulsars (1306)

##  1 Introduction

Globular clusters (GCs) are spherical ensembles of old stars that constitute
as an important part of our Galaxy. They contain $\sim 10^{5}$ stars within
themselves and most of them have typical ages of greater than $10^{10}$ year
(e.g., Harris 1996). Owing to the high stellar densities and thus frequent
dynamical interactions in their cores, GCs are the natural sites to form
compact low-mass X-ray binaries, whose further evolution resulting in
formation of millisecond pulsars (MSPs; Bhattacharya & van den Heuvel 1991).
Therefore GCs can be considered as factories of MSPs. This scenario is
supported by the observational facts that thus far 257 MSPs have been detected
in 36 GCs within $\sim$20 kpc of the Galactic
center111http://www.naic.edu/$\sim$pfreire/GCpsr.html and these GC MSPs
constitute a fraction of $\sim$50% of known MSPs in our Galaxy (Manchester et
al., 2005).

Since the successful launch of the _Fermi Gamma-ray Space Telescope (Fermi)_
and use of the Large Area Telescope (LAT) onboard (Atwood et al., 2009), the
first case of $\gamma$-ray detection of GCs, the 47 Tuc, was reported by (Abdo
et al., 2009), then followed with that of Terzan 5 by Kong et al. (2010). Now
approximately 40 GCs have been reported to have detectable $\gamma$-ray
emission (Abdo et al., 2010b, a; Tam et al., 2011; Zhou et al., 2015; Zhang et
al., 2016; Lloyd et al., 2018a; de Menezes et al., 2019a; Abdollahi et al.,
2020; Yuan et al., 2022). It is generally considered that the $\gamma$-ray
emission of GCs primarily arises from the MSPs contained within them. This
consideration has been well supported by the discoveries of pulsed
$\gamma$-ray emission of an individual pulsar that dominates the detected
$\gamma$-ray emission of a whole GC, specifically PSR J1823$-$3021A in NGC
6624 (Freire et al., 2011) and B1821$-$24 in NGC 6626 (or M28; Wu et al. 2013;
Johnson et al. 2013). Also not only the $\gamma$-ray luminosities of the GCs
but also their $\gamma$-ray spectra have been analyzed to show that the
emission is consistent with that arising from a number of MSPs in each GCs
(Abdo et al., 2010a; Hui et al., 2011; de Menezes et al., 2019b; Lloyd et al.,
2018b; Zhang et al., 2020; Song et al., 2021; Wu et al., 2022).

Table 1: Likelihood analysis results

Models | Parameter values
---|---
LP | $\alpha$ | $\beta$ |  $E_{b}$ (GeV)
| 2.051(98) | 0.400(69) | 1.158(45)
| 2.23(12)⋇ | 0.29(09)⋇ | 1.16⋇
PLEC | $\Gamma$ | $b$ |  $E_{c}$ (GeV)
| 1.544(95) | 1.01(14) | 3.00(33)

⋇4FGL-DR3 values for the LP model, while no errors (numbers in parentheses)
were given for $E_{b}$.

Recently, Gautam et al. (2022) reported the discovery of a 1.83 ms MSP, PSR
J1835$-$3259B, in a near-circular orbit of 28.7 hr within the GC NGC 6652.
This MSP is the second one found in this GC, while the first is PSR
J1835$-$3259A that has a spin period of 3.89 ms and is in a wide binary with
orbital period 9.25 day (DeCesar et al., 2015). NGC 6652 also shows detectable
$\gamma$-ray emission, with the counterpart named J1835.3$-$3255 in the first
Fermi LAT source catalog (1FGL; Abdo et al. 2010a) and latter J1835.7$-$3258
in the fourth catalog (4FGL; Fermi-LAT collaboration et al. 2022). Based on
its $\gamma$-ray spectrum and using a source distance of 10 kpc (Harris,
1996), Wu et al. (2022) estimated a number of 1–7 MSPs in it. Motivated by
these and given the detailed ephemeris for PSR J1835$-$3259B provided in
Gautam et al. (2022), we carried out timing analysis of the Fermi-LAT data for
NGC 6652. We have been able to detect $\gamma$-ray pulsations of PSR
J1835$-$3259B. Here we report the analysis and results.

##  2 Analysis and Results

###  2.1 Fermi-LAT Data and source model

We selected the _Fermi_ -LAT Pass 8 _Front+Back_ events (evclass = 128 and
evtype = 3) in the energy range of 0.1-500.0 GeV within a $20^{\circ}\times
20^{\circ}$ region of interest (RoI) centered at NGC 6652 (R. A. = $\rm
18^{h}35^{m}45\farcs 502$, decl. = $-32^{\circ}58^{{}^{\prime}}15\farcs 621$).
The time range of the data was from 2008 August 04 16:29:16.8 to 2022 June 15
22:12:11.0 (UTC). We excluded the events with zenith angles $>90^{\circ}$ to
avoid the contamination from the Earth’s limb and those with quality flags of
‘bad’ (by the expression DATA_QUAL$>$0 && LAT_CONFIG = 1), and the
instrumental response function ‘P8R3 SOURCE V3’ was used. Thus high-quality
data in good time intervals were selected. In the analysis, the version of
Fermitools–2.0.19 packages were used.

Based on the Data Release 3 of 4FGL (4FGL-DR3; Fermi-LAT collaboration et al.,
2022), which was constructed from 12-yr of LAT data, a model file was made
with the script
make4FGLxml.py.222https://fermi.gsfc.nasa.gov/ssc/data/analysis/user/ The
model file included the spectral parameters of the catalog sources within 25∘
of NGC 6652, and their spectral forms provided in 4FGL-DR3 were used. We set
free the parameters of flux normalizations and spectral shapes for the sources
within 5∘. Other free parameters included the normalizations of the sources
within $5^{\circ}-10^{\circ}$, those outside $10^{\circ}$ but identified with
$Variablility\\_Index\geq 72.44$ (i.e., variable sources), and the Galactic
and extragalactic diffuse emission components. All other parameters were fixed
at their values provided in 4FGL-DR3.

Figure 1: $\gamma$-ray spectrum in 0.1–500 GeV obtained from the data for NGC
6652. The best-fit LP and PLEC models are shown as yellow dashed and blue
solid lines respectively. For the latter, its error range is also shown as the
light blue region. A testing PLEC4 model is shown as the red dash-dotted line
(cf., Section 2.2)

.

###  2.2 Data analysis and spectrum extraction

A binned maximum likelihood analysis was performed to the whole LAT data,
where the log-parabola (LP) model provided in 4FGL-DR3 for NGC 6652,
$\frac{dN}{dE}=N_{0}(\frac{E}{E_{b}})^{-[\alpha+\beta\log(E/E_{b})]}$, was
used. The obtained best-fit parameter values are given in Table 1, which are
in agreement with those given in 4FGL-DR3. The corresponding flux in 0.1–500.0
GeV was $(4.07\pm 0.74)\times 10^{-9}$ photon cm-2 s-1 and test statistic (TS)
value was $\simeq$212\. Because we consider that the $\gamma$-ray emission
likely arises from the MSPs, we then used the model of a power-law with an
exponential cutoff (PLEC),
$\frac{dN}{dE}=N_{0}(\frac{E}{E_{0}})^{-\Gamma}\exp[-(\frac{E}{E_{c}})^{b}]$,
which is typical for describing pulsars’ $\gamma$-ray emission. Performing the
likelihood analysis, the best-fit parameter values were obtained and are given
in Table 1. The corresponding flux in 0.1–500 GeV was $(5.71\pm 0.84)\times
10^{-9}$ photon cm-2 s-1 and TS value was $\sim$208\. As this TS value is only
slightly smaller than that from the LP model, the two models (shown in Figure
1) should be considered equally good for fitting the $\gamma$-ray emission of
NGC 6652. As in 4FGL-DR3, a new parameterization was developed for fitting
emission of pulsars, we tested the model, PLEC4 (see Fermi-LAT collaboration
et al. 2022 for details), and the results were similar to that from PLEC (cf.,
Figure 1).

Figure 2: Timing analysis results for PSR J1835$-$3259B. Panel A: the
integrated weighted $\gamma$-ray pulse profile (16 phase bins). Panel B: two-
dimensional phaseogram (16 phase bins), where the bottom color bar indicates
the weights of the photons (with the largest being 88%). Panel C: H-test
values over the span of the data. Panel D: schematic radio pulse profile of
PSR J1835$-$3259B drawn from that reported in Gautam et al. (2022) for
comparison. Red arrows indicate the mini-peak seen in both $\gamma$-rays and
radio.

We adopted the PLEC model in the following analysis, and the obtained best-fit
parameters were saved in the source model. Using the best-fit parameters of
the PLEC, the integrated energy flux was calculated to be $(4.71\pm
0.39)\times 10^{-12}$ ergs cm-2 s-1, and considering the newly reported source
distance 9.46$\pm 0.14$ kpc for the GC (Baumgardt & Vasiliev, 2021), the
$\gamma$-ray luminosity (assuming isotropic emission) is $L_{\gamma}=(5.04\pm
0.44)\times 10^{34}$ erg s-1.

Based on the updated source model, in which the parameters of spectral shapes
for all the sources were fixed at their best-fit values obtained above and the
normalizations of the sources within 10∘ of the target and the two background
components were set as free parameters, we performed spectral analysis. The
0.1–500.0 GeV energy range was divided into 12 equal logarithmically-spaced
energy bins. A spectrum of the whole LAT data was extracted by performing the
maximum likelihood analysis to the data in each energy bins. The obtained
spectrum is shown in Figure 1, for which we kept the flux data points with TS
values $\geq$4 and showed 95% flux upper limits otherwise. This spectrum is
relatively well described by the PLEC model, whose $\Gamma$ is within and
$E_{c}$ is slightly higher than the respective ranges determined from the
spectra of 104 $\gamma$-ray MSPs (Wu et al., 2022).

###  2.3 Timing Analysis

We selected the events within an aperture radius of 6 degrees (the 68%
containment angle at 0.1
GeV333https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm)
in 0.1–500 GeV band, and assigned weights to them with their probabilities of
originating from the target (using the Fermi tool gtsrcprob). Pulse phases
were assigned to the weighted photons based on the given ephemeris (Gautam et
al., 2022) by employing Tempo2 (Hobbs et al., 2006) with Fermi plug-in (Ray et
al., 2011). The two-dimensional phaseogram and folded pulse profile in 16
phase bins are plotted in Figure 2 (Panel B and A respectively), where the
uncertainties of the counts in each bins were estimated using the method
provided in Abdo et al. (2013). The pulse profile shows a peak at phase
$\sim$0.31–0.44, as the results of the photons with large weights (high
probabilities) mostly located within phase 0.3–0.5, and also a mini-peak at
phase $\sim$0.94–1.0. A high similarity to the shape of the radio pulse
profile (Panel D of Figure 2, which was approximately drawn based on that
reported in Gautam et al. 2022) is thus seen. Following Abdo et al. (2013), we
estimated the background counts (diffuse emission plus contributions from the
nearby sources), and the value was 89.5, equal to the lowest one
(89.3$\pm$2.4) of the pulse bins.

We applied the H-test statistic to the weighted photons (de Jager & Büsching,
2010; Kerr, 2011), and the cumulative H-test value curve over the time span of
the LAT data is shown in Panel C of Figure 2. The H-test value of the whole
data was $\simeq 41$, corresponding to a $p$-value of $7.5\times 10^{-8}$
($\simeq 5.4\sigma$).

###  2.4 Phase-resolved analysis

Given the established pulse profile, we first performed the likelihood
analysis to the data of the P1 and P2 phase ranges shown in Figure 2, for
further investigation of the emissions of the pulse peaks. Using the PLEC
model, the fits to the main pulse peak (P1) and the mini-peak (P2) were
obtained. The best-fit parameters are consistent with those obtained from the
whole data but with large uncertainty ranges, since the TS values were $\simeq
119$ and $\simeq 33$ respectively.

We also performed the analysis to the data of the P1b and Poff phase ranges
(Panel A of Figure 2), for checking the contributions of the major pulsed
emission and the possible off-pulse emission respectively. Likelihood analysis
resulted in TS values of 189 and 1 respectively. The flux upper limit (95%)
for the latter was $\simeq 6.0\times 10^{-10}$ photon cm-2 s-1 (or $\sim
5.3\times 10^{33}$ erg s-1 at the GC’s distance and assuming the PLEC
parameters of the MSP).

##  3 Discussion

By using the ephemeris given by the radio detection of PSR J1835$-$3259B, we
have analyzed the Fermi-LAT data collected for NGC 6652 for approximately 14
years and detected the pulsations of this newly discovered MSP at a
5.4$\sigma$ confidence level. The high similarity of the $\gamma$-ray pulse
profile with the radio one also strongly supports our detection. This pulsar
has a maximum spin-down rate $\dot{P}$ of 6.65$\times 10^{-20}$ (Gautam et
al., 2022), which implies a spin-down luminosity of $\dot{E}\leq 4.3\times
10^{35}$ erg s-1. The $\gamma$-ray efficiency would be
$\eta=L_{\gamma}/\dot{E}\gtrsim 0.12$ when we assign the observed $L_{\gamma}$
to the MSP. The $\eta$ value is in line with those of other MSPs (e.g., Wu et
al. 2022), again supporting the detection.

Wu et al. (2022) have estimated 1–7 MSPs, with 4 MSPs as the likely case, in
NGC 6652. However, the background light curve matches the pulse phase bin of
the lowest counts and the results from the phase-resolved analysis indicate
that the major pulsed part contributes to most of the observed $\gamma$-ray
emission from the GC, strongly suggesting little contributions from other
sources. We note that the first MSP J1835$-$3259A had no reported $\dot{P}$
(set to zero in DeCesar et al. 2015), which thus likely contributes little to
the observed $\gamma$-ray emission. Paduano et al. (2021) reported a candidate
transitional MSP (tMSP) in the GC, which has an X-ray luminosity of $1.8\times
10^{34}$ erg s-1. If this candidate is in a sub-luminous disk state or a
pulsar state, its $\gamma$-ray luminosity would be $\sim 5\times 10^{34}$ erg
s-1 (i.e., $\sim L_{\gamma}$) or much higher respectively (based on the
X-ray–to–$\gamma$-ray flux ratios summarized for tMSPs in Miller et al. 2020).
The $\gamma$-ray detection of J1835$-$3259B actually challenges the
identification of the candidate tMSP. Thus it is very likely that the
$\gamma$-ray emission from the GC comes dominantly from PSR J1835$-$3259B.

Intriguing questions remain to be investigated such as why like NGC 6624 and
NGC 6626, NGC 6652 also contains one exceptionally bright MSP. With
$\gamma$-ray luminosities of $\sim$4–9$\times 10^{34}$erg s-1, the three MSPs
are among the brightest ones (Wu et al., 2022). While the former two have
higher spin-down luminosities of $\sim 10^{36}$ erg s-1, these three all have
relatively high rotational power among all MSPs. Also the former two are
isolated pulsars, but PSR J1835$-$3259B is in a binary likely with a Helium
white dwarf companion (Gautam et al., 2022), along the evolutionary tracks for
such binaries (Tauris & Savonije, 1999). Further studies of pulsars in the GCs
could help our understanding, by pinning down how many such bright MSPs are in
the $\gamma$-ray bright GCs.

We thank anonymous referee for very helpful suggestions and W. Wu for
inspiring this work and helping with the MSP property comparison. This work is
supported in part by the National Key R&D Program of China under grant No.
2018YFA0404204, the National Natural Science Foundation of China No. 12163006,
the Basic Research Program of Yunnan Province No. 202201AT070137, and the
Foundations of Yunnan Province (202201BF070001-020). Z. W. acknowledges the
support by the Original Innovation Program of the Chinese Academy of Sciences
(E085021002) and the Basic Research Program of Yunnan Province No.
202201AS070005.

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# Double calculus

Patrik Lundström  University West, Department of Engineering Science, SE-461
86 Trollhättan, Sweden<EMAIL_ADDRESS>
###### Abstract.

We present a streamlined, slightly modified version, in the two-variable
situation, of a beautiful, but not so well known, theory by Bögel [1, 2],
already from the 1930s, on an alternative higher dimensional calculus of real
functions, a _double calculus_ , which includes two-variable extensions of
many classical results from single variable calculus, such as Rolle’s theorem,
Lagrange’s mean value theorem, Cauchy’s mean value theorem, Fermat’s extremum
theorem, the first derivative test, and the first and second fundamental
theorems of calculus.

###### Key words and phrases:

continuous function, differentiable function, Riemann integral, the
fundamental theorem of calculus

###### 2020 Mathematics Subject Classification:

26A24, 26A42,26B15

##  1\. Introduction

The motivation for this article comes from the trivial observation that the
difference operator $\Delta$ is connected to both the the derivative and the
integral. To be more precise, if $f$ is a single variable real function, then
$f$ has derivative $f^{\prime}(a)=\lim_{b\to a}\Delta_{a}^{b}(f)/(b-a)$ at
$a$, if the limit exists exists, where $\Delta_{a}^{b}(f)$ denotes
$f(b)-f(a)$, and if $f$ is continuous, then, by the fundamental theorem of
calculus, $\int_{a}^{b}f(x)\ dx=\Delta_{a}^{b}(F)$, where $F$ is a primitive
function of $f$. A naive interpretation of this connection is that it should,
in theory, be possible to obtain higher-dimensional analogues of the
fundamental theorem of calculus by first defining a suitable difference
operator given by the multiple integral, over suitable domains, and then, by
reverse engineering, use this difference operator to define a derivative so
that the fundamental theorem of calculus holds.

It may come as a surprise to some readers, and it certainly did so for the
author of the present article, that Bögel [1, 2] already in the 1930s showed
that it is indeed possible, in any finite number of variables, to successfully
carry out such a program. Since this theory should be of high interest to
students and instructors of calculus in several variables, we have in this
article produced an accessible and streamlined, slightly modified version, of
this program in the case of two variables, that is, a theory of _double
calculus_. Note that our notation, definitions, results and proofs, at times,
somewhat differs from the approach by Bögel [1, 2]. In the presentation we
therefore carefully point out whenever that happens.

The domains that Bögel considers are the natural two-dimensional analogues of
intervals, namely _double intervals_ $[a,b]=[a_{1},b_{1}]\times[a_{2},b_{2}]$
where $a=(a_{1},a_{2})$ and $b=(b_{1},b_{2})$ are points in $\mathbb{R}^{2}$
with $a_{1}<b_{1}$ and $a_{2}<b_{2}$. If $f:[a,b]\to\mathbb{R}$ is a
continuous two-variable function, then, by iterated integration, it follows
that $\iint_{[a,b]}f(x)\
dx_{1}dx_{2}=F(b_{1},b_{2})-F(b_{1},a_{2})-F(a_{1},b_{2})+F(a_{1},a_{2})$
where $F$ is a two-variable function with the property that the iterated
partial derivatives $F_{12}$ and $F_{21}$ exist and are equal to $f$ on
$[a,b]$. Therefore he defines the _double difference operator_ by
$\Delta_{a}^{b}(f)=f(b_{1},b_{2})-f(b_{1},a_{2})-f(a_{1},b_{2})+f(a_{1},a_{2})$
and he defines the _double_ _derivative_ of $f$ at $a$ by
$f^{\prime}(a)=\lim_{x_{1}\to a_{1};x_{2}\to
a_{2}}\Delta_{a}^{b}(f)/((b_{1}-a_{1})(b_{2}-a_{2}))$ when it exists.

It turns out that the class of double differentiable functions thus obtained
is much richer than it’s one-dimensional counterpart. In fact, the class of
double differentiable functions contains many examples of functions that are
not partially differentiable and, in some cases, not even continuous. Indeed,
if we pick _any_ functions $g,h:\mathbb{R}\to\mathbb{R}$ and define
$f:\mathbb{R}^{2}\to\mathbb{R}$ by $f(x)=g(x_{1})+h(x_{2})$ for
$x\in\mathbb{R}^{2}$, then $\Delta_{a}^{b}(f)=0$ for all
$a,b\in\mathbb{R}^{2}$ and hence $f$ is _double constant_ , that is $f$ is
double differentiable with $f^{\prime}=0$. This has the unpleasant consequence
that double differentiable functions may be _unbounded_ on compact subsets of
$\mathbb{R}^{2}$. Therefore, we can not expect to find a two-dimensional
version of the Weierstrass extreme value theorem to hold within this
framework.

Here is a detailed outline of this article.

In Section 2, we fix the notation concerning double intervals and double
functions. We also show some elementary results that we need in subsequent
sections.

In Section 3, we define the class of double continuous functions. We show that
a function which is double continuous on an interval is automatically
_globally_ double continuous on that double interval. We also show a
continuity result concerning the double difference map that we need in the
following sections.

In Section 4, we define (signed) double limits and (signed) double
derivatives. In analogue with the single variable situation, we show that
double differentiable functions are double continuous. Then we show double
versions of Rolle’s theorem, the mean value theorem, Fermat’s theorem and the
first derivative test. At the end of this section, we introduce double
primitive functions.

In Section 5, we define the double Newton integral. Using the double mean
value theorem, we obtain a mean value theorem for double Newton integrals.
After that, we connect the double Newton integral to the Riemann double
integral in the first and second double fundamental theorems of calculus. At
the end of this section, we introduce improper double Newton integrals. We
also discuss some examples of double integrals over non-rectangular regions.

In Section 6, we discuss Bögels extensions of the results in this article to
higher dimensions, that is triple calculus, quadruple calculus and beyond. We
also discuss the possibility for a higher-dimensional version of Schwarz’s
theorem and Darboux’s theorem.

##  2\. Double functions

In this section, we fix the notation concerning double intervals and double
functions. We also show some elementary results that we need in subsequent
sections (see Proposition 1 and Proposition 2).

Let $\mathbb{N}$ denote the set of positive integers. We let $\mathbb{R}$
denote the set of real numbers and we put
$\mathbb{R}^{2}:=\mathbb{R}\times\mathbb{R}$. Let $\mathbb{R}_{+}$ and
$\mathbb{R}_{-}$ denote the set of positive real numbers and the set of
negative real numbers respectively; we put
$\mathbb{R}_{++}^{2}:=\mathbb{R}_{+}\times\mathbb{R}_{+}$,
$\mathbb{R}_{+-}^{2}:=\mathbb{R}_{+}\times\mathbb{R}_{-}$,
$\mathbb{R}_{-+}^{2}:=\mathbb{R}_{-}\times\mathbb{R}_{+}$ and
$\mathbb{R}_{--}^{2}:=\mathbb{R}_{-}\times\mathbb{R}_{-}$.

Suppose that $a\in\mathbb{R}^{2}$. We write $a=(a_{1},a_{2})$ for
$a_{1},a_{2}\in\mathbb{R}$. More generally, if $A\subseteq\mathbb{R}^{2}$,
then we put $A_{1}=\\{a_{1}\mid a\in A\\}$ and $A_{2}=\\{a_{2}\mid a\in A\\}$.
Suppose that $b\in\mathbb{R}^{2}$. Then we write $a\sim b$ if $a_{1}=b_{1}$ or
$a_{2}=b_{2}$; $a\nsim b$ if $a_{1}\neq b_{1}$ and $a_{2}\neq b_{2}$; $a<b$ if
$a_{1}<b_{1}$ and $a_{2}<b_{2}$; $a\leq b$ if $a_{1}\leq b_{1}$ and $a_{2}\leq
b_{2}$.

By a _double interval_ we mean a subset $I$ of $\mathbb{R}^{2}$ of the form
$I_{1}\times I_{2}$ where $I_{1}$ and $I_{2}$ are intervals in $\mathbb{R}$.
If $I_{1}$ and $I_{2}$ are open (closed, compact), then we say that $I$ is
_double open (double closed, double compact)_. If $a,b\in\mathbb{R}^{2}$ and
$a\leq b$, then we put $(a,b)=\\{x\in\mathbb{R}^{2}\mid a<x<b\\}$ and
$[a,b]=\\{x\in\mathbb{R}^{2}\mid a\leq x\leq b\\}$.

By a _double $\delta$-neighbourhood of $a$_ we mean a set of the form
$D(a,\delta):=(a-\delta,a+\delta)$ for some $\delta\in\mathbb{R}_{++}^{2}$.
The _signed_ double $\delta$-neighbourhoods $D_{++}(a,\delta)$,
$D_{-+}(a,\delta)$, $D_{--}(a,\delta)$ and $D_{+-}(a,\delta)$ are defined as
the intersection of $D(a,\delta)$ with, respectively, the closure of the
first, the second, the third and the fourth quadrant in $\mathbb{R}^{2}$. By a
_double punctured $\delta$-neighbourhood of $a$_ we mean a set of the form
$P(a,\delta):=\\{x\in D(a,\delta)\mid x\nsim a\\}$ for some
$\delta\in\mathbb{R}_{++}^{2}$. The _signed_ double punctured
$\delta$-neighbourhoods $P_{++}(a,\delta)$, $P_{-+}(a,\delta)$,
$P_{--}(a,\delta)$ and $P_{+-}(a,\delta)$ are defined as the intersection of
$D(a,\delta)$ with, respectively, the first, the second, the third and the
fourth quadrant in $\mathbb{R}^{2}$.

Let $f$ denote a real-valued function with a domain $D(f)$ which is a subset
of $\mathbb{R}^{2}$. In that case, we say that $f$ is a _double function_. If
$a,b\in\mathbb{R}^{2}$ are chosen so that
$(b_{1},b_{2}),(b_{1},a_{2}),(a_{1},b_{2}),(a_{1},a_{2})\in D(f)$, then we
define the _double difference of $f$ from $a$ to $b$_ as the real number:

$\Delta_{a}^{b}(f):=f(b_{1},b_{2})-f(b_{1},a_{2})-f(a_{1},b_{2})+f(a_{1},a_{2}).$

If $I$ is a double interval contained in $D(f)$, then we say that $f$ is
_double constant_ on $I$ if $\Delta_{a}^{b}(f)=0$ for all $a,b\in I$.

######  Proposition 1.

Suppose that $f$ is a double function which is defined on a double interval
$I$. Then $f$ is double constant on $I$ if and only if there are functions
$g:I_{1}\to\mathbb{R}$ and $h:I_{2}\to\mathbb{R}$ with
$f(x)=g(x_{1})+h(x_{2})$ for $x\in I$.

###### Proof.

Suppose $f$ is double constant on $I$. Take $a\in I$. Define
$g:I_{1}\to\mathbb{R}$ by $g(s)=f(s,a_{2})$ for $s\in I_{1}$ and define
$h:I_{2}\to\mathbb{R}$ by $h(t)=f(a_{1},t)-f(a_{1},a_{2})$ for $t\in I_{2}$.
If $x\in I$, then:

$\begin{array}[]{rcl}f(x_{1},x_{2})&=&f(x)-0=f(x_{1},x_{2})-\Delta_{a}^{x}(f)\\\\[5.0pt]
&=&f(x_{1},x_{2})-f(x_{1},x_{2})-f(a_{1},a_{2})+f(x_{1},a_{2})+f(a_{1},x_{2})\\\\[5.0pt]
&=&g(x_{1})+h(x_{2}).\end{array}$

Now suppose that there are functions $g:I_{1}\to\mathbb{R}$ and
$h:I_{2}\to\mathbb{R}$ with $f(x)=g(x_{1})+h(x_{2})$ for $x\in I$. If $a,b\in
I$, then:

$\begin{array}[]{rcl}\Delta_{a}^{b}(f)&=&f(b_{1},b_{2})+f(a_{1},a_{2})-f(b_{1},a_{2})-f(a_{1},b_{2})\\\\[5.0pt]
&=&g(b_{1})+h(b_{2})+g(a_{1})+h(a_{2})\\\\[5.0pt]
&-&\left(g(b_{1})+h(a_{2})+g(a_{1})+h(b_{2})\right)\ =\ 0.\end{array}$

Thus, $f$ is double constant on $I$. ∎

For future reference, we record some properties of the double difference.

######  Proposition 2.

Suppose that $f$ is a double function which is defined on a double interval
containing the points $a$, $b$ and $x$. Then:

  1. (a)

$\Delta_{a}^{a}(f)=0$; $\Delta_{a}^{b}(f)=\Delta_{b}^{a}(f)$;
$\Delta_{a}^{b}(f)=-\Delta_{(a_{1},b_{2})}^{(b_{1},a_{2})}(f)$;

  2. (b)

$\Delta_{a}^{b}(f)=\Delta_{a}^{(x_{1},b_{2})}(f)+\Delta_{(x_{1},a_{2})}^{b}(f)$;

  3. (c)

$\Delta_{a}^{b}(f)=\Delta_{a}^{(b_{1},x_{2})}(f)+\Delta_{(a_{1},x_{2})}^{b}(f)$;

  4. (d)

$\Delta_{a}^{b}(f)=\Delta_{a}^{x}(f)+\Delta_{x}^{b}(f)+\Delta_{(a_{1},x_{2})}^{(x_{1},b_{2})}(f)+\Delta_{(x_{1},a_{2})}^{(b_{1},x_{2})}(f)$.

###### Proof.

These properties follow immediately from the definition of $\Delta$. ∎

##  3\. Double continuity

In this section, we define the class of double continuous functions and the
class of globally double continuous functions. We show that a function which
is double continuous on an interval is automatically _globally_ double
continuous on that double interval (see Proposition 7). We also show a
continuity result (see Proposition 8), concerning the double difference map
that we need in the next section.

Let $a\in\mathbb{R}^{2}$ and suppose that $f$ is a double function. We say
that $f$ is _double continuous at $a$_ if there is a double open interval $I$
with $a\in I\subseteq D(f)$ such that $\forall x_{1}\in I_{1}$ $\lim_{x_{2}\to
a_{2}}\Delta_{a}^{x}(f)=0$ and $\forall x_{2}\in I_{2}$ $\lim_{x_{1}\to
a_{1}}\Delta_{a}^{x}(f)=0$.

Now we define _signed_ double continuity at a point. We say that $f$ is _$+-$
double continuous at $a$_ if there is $\delta\in\mathbb{R}_{++}^{2}$ with
$a\in I:=D_{+-}(a,\delta)\subseteq D(f)$ such that $\forall x_{1}\in I_{1}$
$\lim_{x_{2}\to a_{2}^{-}}\Delta_{a}^{x}(f)=0$ and $\forall x_{2}\in I_{2}$
$\lim_{x_{1}\to a_{1}^{+}}\Delta_{a}^{x}(f)=0$. Analogously, $++$double
continuity, $-+$double continuity and $--$double continuity are defined.

Let $I$ be a double interval such that $I\subseteq D(f)$. If $f$ is double
continuous at every $a\in I$, then we say that $f$ is _double continuous on
$I$_. In this definition it is understood that if $a$ is a boundary point of
$I$, then by double continuity at $a$ we mean this in the sense of the signed
double continuity defined above. Namely, suppose, for instance, that
$I=(a_{1},b_{1}]\times(a_{2},b_{2}]$. By double continuity at the point $b$,
we mean $--$double continuity, and by double continuity at a point
$(b_{1},t)$, where $a_{2}<t<b_{2}$, we mean both $-+$double continuity and
$--$double continuity.

######  Proposition 3.

Suppose that $f$ is a double function. Let $I$ be a double interval.

  1. (a)

If $f$ is continuous on $I$, then $f$ is double continuous on $I$.

  2. (b)

If $f$ is double constant on $I$, then $f$ is double continuous on $I$.

###### Proof.

This is clear. ∎

######  Example 4.

The class of double continuous functions on $\mathbb{R}^{2}$ contains many
examples of functions which are not continuous. Indeed, suppose that $g$ and
$h$ are functions $\mathbb{R}\to\mathbb{R}$ such that $g$ is continuous but
$h$ is not. Put $f(x)=g(x_{1})+h(x_{2})$ for $x\in\mathbb{R}^{2}$. Then,
clearly, $f$ is not continuous. However, by Proposition 1 and Proposition
3(b), $f$ is double continuous.

######  Example 5.

Define the double function $f$ by $f(x)=x_{1}^{x_{2}}\cdot x_{2}^{x_{1}}$, for
$x>0$, and $f(x)=0$, if $x\leq 0$. Note that $f$ is not double constant since
$\Delta_{0}^{x}(f)=f(x)\neq 0$ for all $x>0$. Since:

$f(t,t)=t^{t}\cdot t^{t}=t^{2t}=e^{2t\ln(t)}\to e^{0}=1\neq 0=f(0,0),$

as $t\to 0^{+}$, it follows that $f$ is not continuous at $0$. However, $f$ is
double continuous at $0$. In fact, if $x_{1}>0$, then:

$\Delta_{0}^{x}(f)=f(x)=x_{1}^{x_{2}}\cdot x_{2}^{x_{1}}\to 1\cdot
0=0=f(0,0),$

as $x_{2}\to 0^{+}$; if $x_{2}>0$, then:

$\Delta_{0}^{x}(f)=f(x)=x_{1}^{x_{2}}\cdot x_{2}^{x_{1}}\to 0\cdot
1=0=f(0,0),$

as $x_{1}\to 0^{+}$.

######  Remark 6.

Our definition of double continuity is different from Bögel’s [1] definition
of this concept. Indeed, he defines $f$ to be continuous at $a$ if $\lim_{x\to
a}\Delta_{a}^{x}(f)=0$. Example 5 shows that there are double continuous
functions in our sense that are not double continuous in the sense of Bögel.
On the other hand, the function $f(x)=x_{1}^{2}+x_{2}^{2}$, for $x\nsim 0$,
and $f(x)=0$, for $x\sim 0$, is continuous at 0 in the sense of Bögel but not
in our sense. We have chosen to define double continuity as weak as possible
but so that it still is implied by double differentiability (see Proposition
14).

Suppose that $f$ is a double function. Let $I$ be a double interval such that
$I\subseteq D(f)$. We say that $f$ is globally double continuous on $I$ if
$\forall c,d\in I_{1}$ $\forall e\in I_{2}$ $\lim_{x_{2}\to e;x_{2}\in
I_{2}}\Delta_{(d,e)}^{(c,x_{2})}(f)=0$, and $\forall c,d\in I_{2}$ $\forall
e\in I_{1}$ $\lim_{x_{1}\to e;x_{1}\in I_{1}}\Delta_{(e,d)}^{(x_{1},c)}(f)$
$=0$. To the knowledge of the author of the present article, Bögel [1, 2] does
not define any concept resembling global double continuity.

######  Proposition 7.

Let $I$ be a double interval and suppose that $f$ is a double function such
that $I\subseteq D(f)$. Then $f$ is double continuous on $I$ if and only if
$f$ is globally double continuous on $I$.

###### Proof.

The “if” statement is clear. Now we show the “only if” statement. Suppose that
$f$ is double continuous on $I$. Take $a,b\in I$ with $a<b$. Take
$e\in[a_{2},b_{2}]$ and $c,d\in[a_{1},b_{1}]$ with $c<d$. Put $J=[c,d]$. The
case when $d<c$ is symmetrical and is therefore left to the reader. For all
$z\in J$ choose $\epsilon_{z}>0$ such that for all
$y\in(z-\epsilon_{z},z+\epsilon_{z})$ the map $[a_{2},b_{2}]\ni x_{2}\mapsto
f(y,x_{2})-f(c,x_{2})$ is continuous at $x_{2}=e$. Since the open intervals
$(z-\epsilon_{z}/2,z+\epsilon_{z}/2)$, for $z\in J$, is an open cover of the
compact interval $J$ we can choose a finite subcover
$\\{(z_{i}-\epsilon_{z_{i}}/2,z_{i}+\epsilon_{z_{i}}/2)\\}_{i=0}^{n}$ of $J$.
We may assume that $a_{1}=z_{0}<z_{1}<\cdots<z_{n-1}<z_{n}=x_{1}$. We may also
assume that for all $i,j\in\\{0,\ldots,n\\}$ if
$(z_{i}-\epsilon_{z_{i}}/2,z_{i}+\epsilon_{z_{i}}/2)\subseteq(z_{j}-\epsilon_{z_{j}}/2,z_{j}+\epsilon_{z_{j}}/2)$,
then $i=j$. It follows that:

$\begin{array}[]{rcl}f(x_{1},x_{2})-f(a_{1},x_{2})&=&\sum_{i=1}^{n}f(z_{i},x_{2})-f(z_{i-1},x_{2})\\\\[5.0pt]
&\to&\sum_{i=1}^{n}f(z_{i},a_{2})-f(z_{i-1},a_{2})\\\\[5.0pt]
&=&f(x_{1},a_{2})-f(a_{1},a_{2}),\end{array}$

as $x_{2}\to a_{2}$, since for all $i\in\\{1,\ldots,n\\}$,
$z_{i-1}\in(z_{i}-\epsilon_{z_{i}},z_{i}+\epsilon_{z_{i}})$ or
$z_{i}\in(z_{i-1}-\epsilon_{z_{i-1}},z_{i-1}+\epsilon_{z_{i-1}})$. The
calculation involving the second limit is completely analogous to the above
calculation and is left to the reader. ∎

The next result is [1, Satz 12]. To prove this we use our notion of global
double continuity, whereas Bögel loc. cit. resorts to an ad hoc argument
(although in a more general context).

######  Proposition 8.

Let $a,b,h\in\mathbb{R}^{2}$ where $a<b$. Suppose that $f:[a,b]\to\mathbb{R}$
is a double continuous function. Then the map $x\mapsto\Delta_{x}^{x+h}$ is
continuous at all $x\in(a,b)$ for which $x+h\in(a,b)$.

###### Proof.

It is enough to show that the map $x\mapsto\Delta_{x}^{x+h}$ is continuous
separately in the variables $x_{1}$ and $x_{2}$. By Proposition 7 it follows
that:

$\begin{array}[]{rcl}\Delta_{x}^{x+h}(f)-\Delta_{(a_{1},x_{2})}^{(a_{1}+h_{1},x_{2}+h_{2})}&=&f(x_{1}+h_{1},x_{2}+h_{2})-f(x_{1}+h_{1},x_{2})\\\\[5.0pt]
&&-\ (f(a_{1}+h_{1},x_{2}+h_{2})-f(a_{1}+h_{1},x_{2}))\\\\[5.0pt] &&-\
(f(x_{1},x_{2}+h_{2})-f(x_{1},x_{2}))\\\\[5.0pt] &&+\
f(a_{1},x_{2}+h_{2})-f(a_{1},x_{2})\ \to\ 0,\end{array}$

as $x_{1}\to a_{1}$, and:

$\begin{array}[]{rcl}\Delta_{x}^{x+h}(f)-\Delta_{(x_{1},a_{2})}^{(x_{1}+h_{1},a_{2}+h_{2})}&=&f(x_{1}+h_{1},x_{2}+h_{2})-f(x_{1}+h_{1},x_{2})\\\\[5.0pt]
&&-\ (f(x_{1}+h_{1},a_{2}+h_{2})-f(x_{1}+h_{1},a_{2}))\\\\[5.0pt] &&-\
(f(x_{1},x_{2}+h_{2})-f(x_{1},x_{2}))\\\\[5.0pt] &&+\
f(x_{1},a_{2}+h_{2})-f(x_{1},a_{2})\ \to\ 0,\end{array}$

as $x_{2}\to a_{2}$. ∎

######  Proposition 9.

Let $a,b\in\mathbb{R}^{2}$ where $a<b$. Suppose that $f$ is a double function
which is continuous on $[a,b]$. Let $D$ denote the smallest closed interval in
$\mathbb{R}$ containing $f(a_{1},a_{2})$, $f(b_{1},b_{2})$, $f(a_{1},b_{2})$
and $f(b_{1},a_{2})$. If $d$ is an interior point in $D$, then there exists
$c\in(a,b)$ with $f(c)=d$.

###### Proof.

From the assumptions it follows that there are $a^{\prime},b^{\prime}\in(a,b)$
with $f(a_{1}^{\prime},a_{2}^{\prime})<d<f(b_{1}^{\prime},b_{2}^{\prime})$.
Let $L$ denote the line segment from $a^{\prime}$ to $b^{\prime}$. Since $L$
is connected and $f$ is continuous, $f(L)$ is an interval. Thus, there is
$c\in L$ with $f(c)=d$. Since $c\in L\subsetneq[a,b]$ it follows that
$c\in(a,b)$. ∎

##  4\. Double differentiability

In this section, we define (signed) double limits and (signed) double
derivatives. In analogue with the single variable situation, we show that
double differentiable functions are double continuous (see Proposition 14).
Thereafter, we show a double Rolle’s theorem (see Proposition 17), a double
Lagrange’s mean value theorem (see Proposition 18), a double Cauchy’s mean
value theorem (see Proposition 19), a double Fermat’s theorem (see Proposition
25) and a double first derivative test (see Proposition 26). At the end of
this section, we define double primitive functions.

Suppose that $a\in\mathbb{R}^{2}$, $L\in\mathbb{R}$ and that $f$ is a double
function. We say that $f$ has _double limit $L$ as $x$ approaches $a$_ if for
every $\epsilon>0$ there is $\delta\in\mathbb{R}_{++}^{2}$ with
$P(a,\delta)\subseteq D(f)$ and $|f(x_{1},x_{2})-L|<\epsilon$ whenever $x\in
P(a,\delta)$. In that case, we write $\lim_{x\leadsto a}f(x_{1},x_{2})=L$ or
$f(x_{1},x_{2})\to L$ as $x\leadsto a$.

Now we define _signed_ double limits. We say that $f$ has _$+-$ double limit
$L$ as $x$ approaches $a$_ if for every $\epsilon>0$ there is
$\delta\in\mathbb{R}_{++}^{2}$ with $P_{+-}(a,\delta)\subseteq D(f)$, and
$|f(x_{1},x_{2})-L|<\epsilon$ whenever $x\in P_{+-}(a,\delta)$. In that case,
we write $\lim_{x\leadsto a^{+-}}f(x_{1},x_{2})=L$ or $f(x_{1},x_{2})\to L$ as
$x\leadsto a^{+-}$. Analogously, $++$double limits, $-+$double limits and
$--$double limits are defined.

More generally, one may analogously define (signed) double limits when $a$ and
$L$ belong to
$\overline{\mathbb{R}}^{2}=\overline{\mathbb{R}}\times\overline{\mathbb{R}}$,
where $\overline{\mathbb{R}}$ denotes the affinely extended real number system
$\mathbb{R}\cup\\{\infty,-\infty\\}$. We leave the details of these
definitions to the reader.

Suppose that $a,b\in\mathbb{R}^{2}$ and that $a\nsim b$. If $f$ is defined at
$(a_{1},a_{2})$, $(a_{1},b_{2})$, $(b_{1},a_{2})$ and $(b_{1},b_{2})$, then we
define the _double mean slope of $f$ from $a$ to $b$_ to be the quotient:

$m_{a}^{b}(f):=\frac{\Delta_{a}^{b}(f)}{(b_{1}-a_{1})(b_{2}-a_{2})}.$

We say that $f$ is _double differentiable_ at $a$ if there is a double open
interval $I$ with $a\in I\subseteq D(f)$ and the double limit $\lim_{x\leadsto
a}m_{a}^{x}(f)$ exists. In that case, we let $f^{\prime}(a)$ denote this limit
and we say that $f^{\prime}(a)$ is the _double derivative_ of $f$ at $a$ (cf.
[10]). Using signed double limits, we can analogously define the $++$double
derivative $f^{\prime}_{++}(a)$, the $+-$double derivative
$f^{\prime}_{+-}(a)$, the $-+$double derivative $f^{\prime}_{-+}(a)$ and the
$--$double derivative $f^{\prime}_{--}(a)$.

Let $I$ be a double interval such that $I\subseteq D(f)$. If $f$ is double
differentiable at every $a\in I$, then we say that $f$ is _double
differentiable on $I$_. In this definition it is understood that when $a$ is a
boundary point of $I$, then by double differentiable at $a$ we mean this in
the sense of the signed double differentiability defined above. Namely,
suppose, for instance, that $I=[a_{1},b_{1})\times(a_{2},b_{2}]$. By
differentiability at the point $(a_{1},b_{2})$, we mean $+-$double
differentiability, and by double differentiability at a point $(a_{1},t)$,
where $a_{2}<t<b_{2}$, we mean both $++$double differentiability and
$+-$double differentiability.

The next result, more commonly known as Schwarz’s theorem, Clairaut’s theorem,
or Young’s theorem, is well-known (see e.g. [8, Theorem 9.40]). We have,
nevertheless, chosen to include a proof of it since it involves, in a natural
way, use of the double derivative. Note that for iterated partial derivatives,
we will use the convention that $f_{12}$ and $f_{21}$ denote $(f_{1})_{2}$ and
$(f_{2})_{1}$ respectively.

######  Proposition 10.

Suppose that $f$ is a double function with the property that the mixed partial
derivatives $f_{12}$ and $f_{21}$ exist and are continuous at
$a\in\mathbb{R}^{2}$. Then $f$ is double differentiable at $a$ and
$f^{\prime}(a)=f_{12}(a)=f_{21}(a)$.

###### Proof.

The assumptions imply that $f$ and the partial derivatives $f_{12}$, $f_{21}$,
$f_{1}$ and $f_{2}$ are defined in some double open interval $I$ containing
$a$. Take $h\in\mathbb{R}^{2}$ with $h\nsim 0$. Define functions $u$ and $v$
by:

$u(x_{1})=f(x_{1},a_{2}+h_{2})-f(x_{1},a_{2}),$

for $x_{1}\in I_{1}$, and:

$v(x_{2})=f(a_{1}+h_{1},x_{2})-f(a_{1},x_{2}),$

for $x_{2}\in I_{2}$. Repeated use of the single variable mean value theorem
yield $\theta_{1},\theta_{2}\in(0,1)$ such that:

$\displaystyle\Delta_{a}^{a+h}(f)$ $\displaystyle=$ $\displaystyle
u(a_{1}+h_{1})-u(a_{1})=h_{1}u^{\prime}(a_{1}+\theta_{1}h_{1})$
$\displaystyle=$ $\displaystyle
h_{1}(f_{1}(a_{1}+\theta_{1}h_{1},a_{2}+h_{2})-f_{1}(a_{1}+\theta_{1}h_{1},a_{2}))$
$\displaystyle=$ $\displaystyle
h_{1}h_{2}f_{12}(a_{1}+\theta_{1}h_{1},a_{2}+\theta_{2}h_{2}).$

Thus, $m_{a}^{a+h}(f)=f_{12}(a_{1}+\theta_{1}h_{1},a_{2}+\theta_{2}h_{2})\to
f_{12}(a_{1},a_{2})$ as $h\leadsto 0$. Similarly, there exist
$\theta_{3},\theta_{4}\in(0,1)$ with:

$\displaystyle\Delta_{a}^{a+h}(f)$ $\displaystyle=$ $\displaystyle
v(a_{2}+h_{2})-v(a_{2})=h_{3}v^{\prime}(a_{2}+\theta_{3}h_{2})$
$\displaystyle=$ $\displaystyle
h_{3}(f_{2}(a_{1}+h_{1},a_{2}+\theta_{3}h_{2})-f_{2}(a_{1},a_{2}+\theta_{3}h_{2}))$
$\displaystyle=$ $\displaystyle
h_{1}h_{2}f_{21}(a_{1}+\theta_{4}h_{1},a_{2}+\theta_{3}h_{2}).$

Hence, $m_{a}^{a+h}(f)=f_{21}(a_{1}+\theta_{4}h_{1},a_{2}+\theta_{3}h_{2})\to
f_{21}(a_{1},a_{2})$ as $h\leadsto 0$. Thus, $f$ is double differentiable at
$a$ and $f^{\prime}(a)=f_{12}(a)=f_{21}(a)$. ∎

######  Example 11.

By Proposition 10, all sufficiently smooth double functions are double
differentiable. However, the class of double differentiable functions contains
examples of functions which are not even partially differentiable. In fact, it
is clear that all double constant functions are double differentiable with
double derivative equal to zero everywhere. Therefore, the class of double
differentiable functions even contains many examples of everywhere
discontinuous functions (see Example 4).

######  Example 12.

It is easy to construct examples of double functions that are not double
constant but double differentiable at a point where an iterated partial
derivative fails to exist. Namely, suppose that $g$ and $h$ are single
variable functions defined on $\mathbb{R}$. Furthermore, suppose that
$g(x_{1})\to 0$ as $x_{1}\to 0$; $g(x_{1})\neq 0$ for non-zero
$x_{1}\in\mathbb{R}$; $h(x_{2})$ is bounded near the origin; in any open
interval around the origin, there exists $x_{2}$ such that $h(x_{2})\neq 0$;
$\lim_{x_{2}\to 0}h(x_{2})$ does not exist. For instance, we can choose
$g(x_{1})=x_{1}$ and $h(x_{2})=\sin(1/x_{2})$, for $x_{2}\neq 0$, and
$h(0)=0$. Now put $f(x)=x_{1}g(x_{1})\cdot x_{2}h(x_{2})$ for
$x\in\mathbb{R}^{2}$. Then, in any double open interval containing the origin,
there exists $x$ such that $\Delta_{0}^{x}(f)=f(x)\neq 0$. Also, clearly, the
double derivative $f^{\prime}(0,0)$ exists and is equal to zero. However, for
any fixed nonzero $x_{1}\in\mathbb{R}$, $f_{2}(x_{1},0)=\lim_{x_{2}\to
0}f(x)/x_{2}=\lim_{x_{2}\to 0}x_{1}g(x_{1})h(x_{2})$ does not exist. Thus, the
iterated partial derivative $f_{21}(0,0)$ does not exist.

In analogy with the single variable situation, there is a “first order”
approximation to $\Delta$:

######  Proposition 13.

Suppose that $f$ is a double function and that $a\in\mathbb{R}^{2}$. Then $f$
is double differentiable at $a$ if and only if there is $d\in\mathbb{R}$, a
double neighbourhood $D(a,\delta)$, where $f$ is defined, and a function
$\rho:P(a,\delta)\to\mathbb{R}$ such that
$\Delta_{a}^{x}(f)=(x_{1}-a_{1})(x_{2}-a_{2})d+(x_{1}-a_{1})(x_{2}-a_{2})\rho(x)$,
for $x\in P(a,\delta)$, and $\lim_{x\leadsto a}\rho(x)=0$. In that case,
$f^{\prime}(a)=d$.

###### Proof.

This follows immediately from the definition of $\Delta$. ∎

######  Proposition 14.

Suppose that $f$ is a double function which is defined at
$a\in\mathbb{R}^{2}$. If $f$ is double differentiable at $a$, then $f$ is
double continuous at $a$.

###### Proof.

Suppose that $f$ is double differentiable at $a$. Take $\delta$ and $\rho$
satisfying the conditions in the formulation of Proposition 13. If $x\nsim a$,
then:

$\begin{array}[]{rcl}f(x_{1},x_{2})-f(a_{1},x_{2})&=&\Delta_{a}^{x}(f)+f(x_{1},a_{2})-f(a_{1},a_{2})\\\\[5.0pt]
&=&(x_{1}-a_{1})(x_{2}-a_{2})d+(x_{1}-a_{1})(x_{2}-a_{2})\rho(x)\\\\[5.0pt]
&+&f(x_{1},a_{2})-f(a_{1},a_{2})\ \to\
f(x_{1},a_{2})-f(a_{1},a_{2}),\end{array}$

as $x_{2}\to a_{2}$, and:

$\begin{array}[]{rcl}f(x_{1},x_{2})-f(x_{1},a_{2})&=&\Delta_{a}^{x}(f)+f(a_{1},x_{2})-f(a_{1},a_{2})\\\\[5.0pt]
&=&(x_{1}-a_{1})(x_{2}-a_{2})d+(x_{1}-a_{1})(x_{2}-a_{2})\rho(x)\\\\[5.0pt]
&+&f(a_{1},x_{2})-f(a_{1},a_{2})\ \to\
f(a_{1},x_{2})-f(a_{1},a_{2}),\end{array}$

as $x_{1}\to a_{1}$. ∎

We now proceed to show double versions of Rolle’s theorem and various versions
of the mean value theorem. To this end, we need two propositions. We adapt, to
the situation at hand, an approach which originally was invented by Cauchy
(see [5, p. 169]) for the single variable situation and then corrected and
clarified by Plante [7].

######  Proposition 15.

Suppose that $a,b\in\mathbb{R}^{2}$, $a<b$, and that $x\in(a,b)$. Let
$f:[a,b]\to\mathbb{R}$ be a double function. Consider the following four real
numbers:

$m_{1}:=m_{a}^{x}(f),\ m_{2}:=m_{x}^{b}(f),\
m_{3}:=\frac{\Delta_{(a_{1},x_{2})}^{(x_{1},b_{2})}(f)}{(x_{1}-a_{1})(b_{2}-x_{2})},\
m_{4}:=\frac{\Delta_{(x_{1},a_{2})}^{(b_{1},x_{2})}(f)}{(b_{1}-x_{1})(x_{2}-a_{2})}.$

Then either (i) all of them are equal to $m_{a}^{b}(f)$, or (ii) at least one
of them is greater than $m_{a}^{b}$ and at least one of them is less than
$m_{a}^{b}$.

###### Proof.

Suppose that (i) does not hold. Seeking a contradiction, suppose that
$m_{a}^{b}(f)\leq m_{i}$ for all $i\in\\{1,2,3,4\\}$ with strict inequality
for at least one index. By Proposition 2(d), we get that:

$\begin{array}[]{rcl}\Delta_{a}^{b}(f)&=&m_{a}^{b}(f)(b_{1}-a_{1})(b_{2}-a_{2})\\\\[5.0pt]
&=&m_{a}^{b}(f)(x_{1}-a_{1})(x_{2}-a_{2})+m_{a}^{b}(f)(b_{1}-x_{1})(b_{2}-x_{2})\\\\[5.0pt]
&+&m_{a}^{b}(f)(x_{1}-a_{1})(b_{2}-x_{2})+m_{a}^{b}(f)(b_{1}-x_{1})(x_{2}-a_{2})\\\\[5.0pt]
&<&m_{1}(x_{1}-a_{1})(x_{2}-a_{2})+m_{2}(b_{1}-x_{1})(b_{2}-x_{2})\\\\[5.0pt]
&+&m_{3}(x_{1}-a_{1})(b_{2}-x_{2})+m_{4}(b_{1}-x_{1})(x_{2}-a_{2})\\\\[5.0pt]
&=&\Delta_{a}^{x}(f)+\Delta_{x}^{b}(f)+\Delta_{(a_{1},x_{2})}^{(x_{1},b_{2})}(f)+\Delta_{(x_{1},a_{2})}^{(b_{1},x_{2})}(f)\
=\ \Delta_{a}^{b}(f)\end{array}$

and hence the contradiction $\Delta_{a}^{b}(f)<\Delta_{a}^{b}(f)$. Similarly,
if we assume that $m_{a}^{b}(f)\geq m_{i}$, for all $i\in\\{1,2,3,4\\}$, with
strict inequality for at least one index, then we get the contradiction
$\Delta_{a}^{b}(f)>\Delta_{a}^{b}(f)$. Thus (ii) holds. ∎

######  Proposition 16.

Suppose that $a,b\in\mathbb{R}^{2}$ satisfy $a<b$. Let $f:(a,b)\to\mathbb{R}$
be a double function which is double differentiable at $c\in(a,b)$. Let
$a(1)\leq a(2)\leq a(3)\leq\cdots$ and $b(1)\geq b(2)\geq b(3)\geq\cdots$ be
sequences in $(a,b)$ with $\lim_{n\to\infty}a(n)=\lim_{n\to\infty}b(n)=c$ and
satisfying one of the following properties:

  * (i) $\forall n\in\mathbb{N}$ $a(n)<c<b(n)$;

  * (ii) $\forall n\in\mathbb{N}$ $a(n)<c$ and $\exists N\in\mathbb{N}$ $\forall n\geq N$ $c=b(n)$;

  * (iii) $\forall n\in\mathbb{N}$ $c<b(n)$ and $\exists N\in\mathbb{N}$ $\forall n\geq N$ $a(n)=c$.

Then $\lim_{n\to\infty}m_{a(n)}^{b(n)}(f)=f^{\prime}(c)$.

###### Proof.

Suppose that (i) holds. For every $n\in\mathbb{N}$, put:

$m_{1}(n):=\frac{\Delta_{a(n)}^{c}(f)}{(c_{1}-a_{1}(n))(c_{2}-a_{2}(n))}\quad
m_{2}(n):=\frac{\Delta_{c}^{b(n)}(f)}{(b_{1}(n)-c_{1})(b_{2}(n)-c_{2})}$
$m_{3}(n):=\frac{\Delta_{(a_{1}(n),c_{2})}^{(c_{1},b_{2}(n))}(f)}{(c_{1}-a_{1}(n))(b_{2}(n)-c_{2})}\quad
m_{4}(n):=\frac{\Delta_{(c_{1},a_{2}(n))}^{(b_{1}(n),c_{2})}(f)}{(b_{1}(n)-c_{1})(c_{2}-a_{2}(n))}.$

From the definition of the double derivative we get that
$\lim_{n\to\infty}m_{i}(n)=f^{\prime}(c)$ for all $i\in\\{1,2,3,4\\}$. Thus,
from Proposition 15, it follows that
$\lim_{n\to\infty}m_{a(n)}^{b(n)}(f)=f^{\prime}(c)$.

If (ii) (or (iii)) holds, then, from the definition of the double derivative,
we get that
$\lim_{n\to\infty}m_{a(n)}^{b(n)}(f)=\lim_{n\to\infty}m_{a(n)}^{c}(f)=f^{\prime}(c)$
(or
$\lim_{n\to\infty}m_{a(n)}^{b(n)}(f)=\lim_{n\to\infty}m_{c}^{b(n)}(f)=f^{\prime}(c)$).
∎

The next result is [1, Satz 13]. Noteworthily, Bögel proves this result by an
argument which is not identical, but similar in spirit, to our proof. Thus, in
particular, for single variable functions, Bögel’s proof produces the
classical Rolle’s theorem without resorting to the Weierstrass extremum
theorem (cf. [7]).

######  Proposition 17 (Double Rolle’s theorem).

Let $a,b\in\mathbb{R}^{2}$ satisfy $a<b$. Let $f:[a,b]\to\mathbb{R}$ be a
double continuous function which is double differentiable on $(a,b)$. If
$\Delta_{a}^{b}(f)=0$, then there exists $c\in(a,b)$ with $f^{\prime}(c)=0$.

###### Proof.

We claim that there are $p,q\in(a,b)$ with $\Delta_{p}^{q}(f)=0$, $p<q$ and
$q-p\leq(b-a)/2$. Let us assume for a moment that the claim holds. Then we can
inductively define sequences $a(1)\leq a(2)\leq a(3)\leq\cdots$ and $b(1)\geq
b(2)\geq b(3)\geq\cdots$ in $(a,b)$ satisfying $a(n)<b(n)$,
$\Delta_{a(n)}^{b(n)}=0$ and $b(n)-a(n)\leq(b-a)/2^{n}$, for every
$n\in\mathbb{N}$. Then $\\{a(n)\\}_{n=1}^{\infty}$ and
$\\{b(n)\\}_{n=1}^{\infty}$ have a common limit $c\in(a,b)$ satisfying one of
the properties (i)-(iii) in Proposition 16. Thus, from the same proposition,
it follows that $f^{\prime}(c)=0$.

Now we show the claim. Seeking a contradiction, suppose that
$\Delta_{p}^{q}(f)\neq 0$ for all $p,q\in(a,b)$ such that $p<q$ and
$q-p\leq(b-a)/2$. Put $h=(b-a)/2$.

_Case 1: $\Delta_{a}^{a+h}(f)\neq 0$_. Consider the map
$g(x)=\Delta_{x}^{x+h}(f)$ for $x\in[a,a+h]$. From Proposition 2 it follows
that
$g(a)+g(a_{1},(b_{2}+a_{2})/2)+g((b_{1}+a_{1})/2,a_{2})+g((a+b)/2)=\Delta_{a}^{b}(f)=0$.
Therefore, since $g(a)=\Delta_{a}^{a+h}(f)\neq 0$, at least one the real
numbers $g(a)$, $g(a_{1},h_{2}/2)$, $g((b_{1}+a_{1})/2,a_{2})$ and
$g((a+b)/2)$ is positive and at least one is negative. By Proposition 8 and
Proposition 9 it follows that there is $p\in(a,(a+b)/2)$ with $g(p)=0$. Put
$q=p+h=p+(b-a)/2$. Then $\Delta_{p}^{q}(f)=g(p)=0$, $p,q\in(a,b)$, $p<q$ and
$q-p\leq(b-a)/2$. This is a contradiction.

_Case 2: $\Delta_{a}^{a+h}(f)=0$_. Put $b^{\prime}=a+h$ and $h^{\prime}=h/2$.
Then $\Delta_{a}^{b^{\prime}}(f)=0$ and, by the assumptions, we get that
$\Delta_{a+h^{\prime}}^{b^{\prime}}(f)\neq 0$. Consider the map
$g(x)=\Delta_{x}^{x+h^{\prime}}(f)$ for $x\in[a,a+h^{\prime}]$. From
Proposition 2 it follows that
$g(a)+g(a_{1},(b_{2}+a_{2})/4)+g((b_{1}+a_{1})/4,a_{2})+g((a+b)/4)=\Delta_{a}^{b^{\prime}}(f)=0$.
Therefore, since $g(a)=\Delta_{a+h^{\prime}}^{b^{\prime}}(f)\neq 0$, at least
one the real numbers $g(a)$, $g(a_{1},(b_{2}+a_{2})/4)$,
$g((b_{1}+a_{1})/4,a_{2})$ and $g((a+b)/4)$ is positive and at least one is
negative. By Proposition 8 and Proposition 9 it follows that there is
$p^{\prime}\in(a,(a+b)/4)$ with $g(p^{\prime})=0$. Put
$q^{\prime}=p^{\prime}+h^{\prime}=p^{\prime}+(b-a)/4$. Then
$\Delta_{p^{\prime}}^{q^{\prime}}(f)=g(p^{\prime})=0$,
$p^{\prime},q^{\prime}\in(a,b)$, $p^{\prime}<q^{\prime}$ and
$q^{\prime}-p^{\prime}\leq(b-a)/4$. This is a contradiction. ∎

######  Proposition 18 (Double Lagrange’s mean value theorem).

Let $a,b\in\mathbb{R}^{2}$ satisfy $a<b$. Suppose that $f:[a,b]\to\mathbb{R}$
is a double continuous function which is double differentiable on $(a,b)$.
Then there exists $c\in(a,b)$ with $f^{\prime}(c)=m_{a}^{b}(f).$

###### Proof.

Consider the map $g(x)=f(x)-m_{a}^{b}(f)(x_{1}-a_{1})(x_{2}-a_{2})$ for
$x\in[a,b]$. Then it is clear that
$\Delta_{a}^{b}(g)=\Delta_{a}^{b}(f)-\Delta_{a}^{b}(f)=0$. Proposition 17
implies the existence of an element $c\in(a,b)$ with $g^{\prime}(c)=0$. From
Proposition 10 it follows that
$0=g^{\prime}(c)=f^{\prime}(c)-m_{a}^{b}(f)\cdot 1$. Therefore
$f^{\prime}(c)=m_{a}^{b}(f)$. ∎

######  Proposition 19 (Double Cauchy’s mean value theorem [1, Satz 14]).

Let $a,b\in\mathbb{R}^{2}$ satisfy $a<b$. Suppose that
$f,g:[a,b]\to\mathbb{R}$ are double continuous functions which are double
differentiable on $(a,b)$. Then there exists $c\in(a,b)$ with
$f^{\prime}(c)\Delta_{a}^{b}(g)=g^{\prime}(c)\Delta_{a}^{b}(f).$

###### Proof.

We consider two cases.

_Case 1: $\Delta_{a}^{b}(g)=0$._ By Proposition 17 there is $c\in(a,b)$ with
$g^{\prime}(c)=0$. For this $c$ we get that
$f^{\prime}(c)\Delta_{a}^{b}(g)=f^{\prime}(c)\cdot
0=0\cdot\Delta_{a}^{b}(f)=g^{\prime}(c)\Delta_{a}^{b}(f).$

_Case 2: $\Delta_{a}^{b}(g)\neq 0$._ Consider the map
$h(x)=f(x)-\Delta_{a}^{b}(f)g(x)/\Delta_{a}^{b}(g)$ for $x\in[a,b]$. Then,
clearly, $\Delta_{a}^{b}(h)=0$. By Proposition 17 there is $c\in(a,b)$ with
$h^{\prime}(c)=0$ form which the claim follows. ∎

######  Remark 20.

(a) If we specialize $g(x)=x_{1}x_{2}$ in Proposition 19, then we get
Proposition 18.

(b) In [4] the concept of double derivative (there named _bidimensional
derivative_) as well as Propositions 17-19 were rediscovered (independently
from Bögel it seems) by Dobrescu and Siclovan.

Suppose that $f$ is a double function defined on a double interval $I$. We say
that $f$ is _double increasing (double decreasing)_ on $I$ if
$\Delta_{a}^{b}(f)>0$ ($\Delta_{a}^{b}(f)<0$) for all $a,b\in I$ with $a<b$.

######  Example 21.

Let $a,b,c\in\mathbb{R}^{2}$ and $D\in\mathbb{R}$. Define a double function
$f$ on $\mathbb{R}^{2}$ by $f(x)=D(x_{1}-c_{1})(x_{2}-c_{2})$ for
$x\in\mathbb{R}^{2}$. Then:

$\begin{array}[]{rcl}\Delta_{a}^{b}(f)&=&D(b_{1}-c_{1})(b_{2}-c_{2})-D(b_{1}-c_{1})(a_{2}-c_{2})\\\\[5.0pt]
&-&D(a_{1}-c_{1})(b_{2}-c_{2})+D(a_{1}-c_{1})(a_{2}-c_{2})\\\\[5.0pt]
&=&D(b_{1}-a_{1})(b_{2}-a_{2}).\end{array}$

Thus, $f$ is double increasing on $\mathbb{R}^{2}$ $\Leftrightarrow$ $D>0$;
$f$ is double decreasing on $\mathbb{R}^{2}$ $\Leftrightarrow$ $D<0$; $f$ is
double constant on $\mathbb{R}^{2}$ $\Leftrightarrow$ $D=0$.

######  Proposition 22.

Suppose that $a,b\in\mathbb{R}^{2}$ satisfy $a<b$. Let $f:[a,b]\to\mathbb{R}$
be a double continuous function which is double differentiable on $(a,b)$.
Then:

  1. (a)

$f$ is double increasing on $[a,b]$ $\Leftrightarrow$ $f^{\prime}(x)>0$ for
every $x\in(a,b)$;

  2. (b)

$f$ is double decreasing on $[a,b]$ $\Leftrightarrow$ $f^{\prime}(x)<0$ for
every $x\in(a,b)$;

  3. (c)

$f$ is double constant on $[a,b]$ $\Leftrightarrow$ $f^{\prime}(x)=0$ for
every $x\in(a,b)$.

###### Proof.

This follows immediately from Proposition 18. ∎

######  Remark 23.

Consider the function $f$ defined in Example 21. Then, by Proposition 10, it
follows that $f$ is double differentiable with $f^{\prime}(x)=D$ for
$x\in\mathbb{R}^{2}$. Thus, the conclusions in this example follow from
Proposition 22.

Suppose that $f$ is a double function defined on a double open interval $I$.
Let $a\in I$. We say that $a$ is a _double maximum point (double minumum
point)_ for $f$ on $I$ if $\Delta_{a}^{b}(f)<0$ ($\Delta_{a}^{b}(f)>0$) for
all $b\in I$ with $b\nsim a$. We say that $a$ is a _double extreme point_ for
$f$ on $I$ if $a$ is a double minumum point or a double maximum point for $f$
on $I$

######  Example 24.

Let $a,b,c\in\mathbb{R}^{2}$ and $D\in\mathbb{R}$. Define a double function
$f$ on $\mathbb{R}^{2}$ by $f(x)=D(x_{1}-c_{1})^{2}(x_{2}-c_{2})^{2}$ for
$x\in\mathbb{R}^{2}$ where $D$ is a non-zero real number. Then:

$\begin{array}[]{rcl}\Delta_{a}^{b}(f)&=&D(b_{1}-c_{1})^{2}(b_{2}-c_{2})^{2}-D(b_{1}-c_{1})^{2}(a_{2}-c_{2})^{2}\\\\[5.0pt]
&-&D(a_{1}-c_{1})^{2}(b_{2}-c_{2})^{2}+D(a_{1}-c_{1})^{2}(a_{2}-c_{2})^{2}\\\\[5.0pt]
&=&D(b_{1}-a_{1})(b_{2}-a_{2})(b_{1}+a_{1}-2c_{2})(b_{2}+a_{2}-2c_{2}).\end{array}$

Suppose now that $a$ is a double extreme point for $f$ on $\mathbb{R}^{2}$.
The above calculation shows that $\Delta_{a}^{2c-a}=0$. Therefore $a=2c-a$ and
thus $a=c$. Hence $\Delta_{a}^{b}(f)=D(b_{1}-a_{1})^{2}(b_{2}-a_{2})^{2}$.
Thus, $a$ is a double maximum (minimum) point for $f$ on $\mathbb{R}^{2}$ if
and only if $a=c$ and $D<0$ ($D>0$).

Suppose that $f$ is a double function which is defined on a double open
interval $I$ and let $a\in I$. If $f$ is double differentiable at $a$ and
$f^{\prime}(a)=0$, then $f$ we say that $a$ is a _double stationary point_ for
$f$. We say that $a$ is a _double a critical point for $f$_ if either $f$ is
not double differentiable at $a$ or $a$ is a stationary point for $f$.

######  Proposition 25 (Double Fermat’s theorem).

Suppose that $f$ is a double function which is defined on a double open
interval $I$ and let $a\in I$. If $a$ is a double extreme point for $f$ on
$I$, then $a$ is a double critical point for $f$.

###### Proof.

Suppose that $f$ is double differentiable at $a$ and that $a$ is a double
minimum point for $f$ on $I$. Then $\Delta_{a}^{b}(f)>0$ for all $b$ with
$b\nsim a$. Hence:

$f^{\prime}(a)\quad=\lim_{\quad b\leadsto a^{++}}m_{a}^{b}(f)\quad=\lim_{\quad
b\leadsto a^{++}}\frac{\Delta_{a}^{b}(f)}{(b_{1}-a_{1})(b_{2}-a_{2})}\geq 0$

and:

$f^{\prime}(a)\quad=\lim_{\quad b\leadsto a^{+-}}m_{a}^{b}(f)\quad=\lim_{\quad
b\leadsto a^{+-}}\frac{\Delta_{a}^{b}(f)}{(b_{1}-a_{1})(b_{2}-a_{2})}\leq 0.$

Thus $f^{\prime}(a)=0$. The proof is analogous in the case when $f$ is double
differentiable at $a$ and $a$ is a double maximum point for $f$ on $I$. ∎

######  Proposition 26 (Double first derivative test).

Suppose that $f$ is a double function which is double differentiable on a
double open interval $(a,b)$. Let $c\in(a,b)$ be a double stationary point for
$f$.

  1. (a)

Suppose that $f^{\prime}(x)<0$, when $a<x<c$ or $c<x<b$, and $f^{\prime}(x)>0$
when $(a_{1},c_{2})<x<(c_{1},b_{2})$ or $(c_{1},a_{2})<x<(b_{1},c_{2})$. Then
$c$ is a double maximum point for $f$ on $(a,b)$.

  2. (b)

Suppose that $f^{\prime}(x)>0$, when $a<x<c$ or $c<x<b$, and $f^{\prime}(x)<0$
when $(a_{1},c_{2})<x<(c_{1},b_{2})$ or $(c_{1},a_{2})<x<(b_{1},c_{2})$. Then
$c$ is a double minimum point for $f$ on $(a,b)$.

  3. (c)

If $f^{\prime}$ has the same sign throughout the formulation of the statement
in (a) (or in (b)), then $c$ is neither a double maximum point nor a double
minimum point for $f$ on $(a,b)$.

###### Proof.

(a) Take $x\in(a,b)$ with $x\nsim c$. By Proposition 22(b) it follows that $f$
is double decreasing on the intervals $(a,c)$ and $(c,b)$. Thus
$\Delta_{x}^{c}(f)<0$ for all $x$ in those intervals. By Proposition 22(a) it
follows that $f$ is double increasing on the intervals
$((a_{1},c_{2}),(c_{1},b_{2}))$ and $((c_{1},a_{2}),(b_{1},c_{2}))$. By Lemma
2(c) it follows that $\Delta_{x}^{c}(f)>0$ for all $x$ in those intervals. The
proofs of (b) and (c) are similar to the proof of (a). ∎

######  Remark 27.

One can reach the conclusion in Example 24, using the double derivative.
Namely, let $a$ $b$, $c$, $D$ and $f$ be defined as in that example. By
Proposition 10, it follows that $f$ is double differentiable with
$f^{\prime}(x)=4D(x_{1}-c_{1})(x_{2}-c_{2})$ for $x\in\mathbb{R}^{2}$.

Suppose that $a$ is a double maximum point for $f$ on $\mathbb{R}^{2}$. By
Proposition 25 we get that $f^{\prime}(a)=0$, that is
$4D(a_{1}-c_{1})(a_{1}-c_{1})=0$. Therefore $a_{1}=c_{1}$ or $a_{2}=c_{2}$. We
consider the case when $a_{1}=c_{1}$ (the case when $a_{2}=c_{2}$ reaches the
same conclusion) so that $f^{\prime}(x)=4D(x_{1}-a_{1})(x_{2}-c_{2})$. By
Proposition 26 $f^{\prime}(x)<0$ for large enough $x$. Thus $D<0$. Also, by
the same proposition, if $x_{1}\neq a_{1}$, then $f^{\prime}(x)$ should change
sign as $x_{2}$ goes from a value less than $a_{2}$ to a value larger than
$a_{2}$. Hence $c_{2}=a_{2}$ so that
$f^{\prime}(x)=4D(x_{1}-a_{1})(x_{2}-a_{2})$. Using this and Proposition 26,
it is easy to see that $a$ now is a double maximum point for $f$ on
$\mathbb{R}^{2}$.

A similar analysis reveals that $a$ is double minimum point for $f$ on
$\mathbb{R}^{2}$ if and only if $a=c$ and $D>0$.

######  Remark 28.

To the best of our knowledge, Bögel neither treats double critical points nor
a double Fermat’s theorem. However, in [2, §6] Bögel studies _monotone_ double
functions in the context of functions of bounded variation.

Suppose that $f$ is a double function defined on a double interval $I$. We say
that a double function $F$ defined on $I$ is a _double primitive function of
$f$ on $I$_ if $F$ is double differentiable on $I$ and $F^{\prime}(x)=f(x)$
for $x\in I$.

######  Proposition 29.

Suppose that $f$ is a double function defined on a double interval $I$. If $F$
and $G$ are double primitive functions of $f$ on $I$, then there is a double
constant function $H$, defined on $I$, such that $G=F+H$.

###### Proof.

Put $H=G-F$. Then $H^{\prime}=G^{\prime}-F^{\prime}=f-f=0$. Proposition 22(c)
implies that $H$ is a double constant function. Clearly $G=F+H$. ∎

##  5\. Double integrability

In this section, we define the double Newton integral. Using the double mean
value theorem, we obtain a mean value theorem for double Newton integrals (see
Proposition 34). After that, we connect the double Newton integral to the
Riemann double integral in the first and second double fundamental theorems of
calculus (see Proposition 36 and Proposition 37). At the end of this section,
we introduce improper double Newton integrals. We also discuss examples of
double integrals over non-rectangular regions. Most of the material in this
section (except the discussion on improper integrals) can be extracted and
specialized from [2]. However, since we only restrict ourselves to the double
calculus, our presentation can be more streamlined.

Let $f$ be a double function defined on a double interval $I$. Suppose that
there exists a double primitive function $F$ of $f$ on $I$. Let $a,b\in I$. We
say that the _double Newton integral of $f$ from $a$ to $b$_ is the real
number:

(1) $\int_{a}^{b}f:=\Delta_{a}^{b}(F).$

######  Proposition 30 (The double Newton integral is well defined).

The value of (1) does not depend on the choice of the double primitive
function.

###### Proof.

Let $f,F,G$ be double functions defined on $I$ where $F$ and $G$ are double
primitive functions of $f$ on $I$. By Proposition 29, $G=F+H$ for some double
constant function $H$ defined on $I$. Take $a,b\in I$. Then it follows that
$\Delta_{a}^{b}(G)=\Delta_{a}^{b}(F)+\Delta_{a}^{b}(H)=\Delta_{a}^{b}(F)+0=\Delta_{a}^{b}(F).$
∎

######  Example 31.

Suppose that $F$ and $f$ are the double functions defined on
$I:=[0,2]\times[1,3]$ by $F(x)=x_{1}^{2}x_{2}^{3}/2$ and
$f(x)=3x_{1}x_{2}^{2}$ for $x\in I$. By Proposition 10, $F$ is double
differentiable on $I$ with $F^{\prime}=f$. Therefore:

$\displaystyle{\int_{(0,1)}^{(2,3)}f}\ =\ \Delta_{(0,1)}^{(2,3)}(F)\ =\
F(2,3)-F(2,1)-F(0,3)+F(0,1)\ =\ 52.$

######  Proposition 32 (Properties of the double Newton integral).

Let $f$ be a double function defined on a double interval $I$. Suppose that
there exists a double primitive function $F$ of $f$ on $I$. If $a,b,c\in I$,
then:

  1. (a)

$\displaystyle{\int_{a}^{a}f=0}$;
$\displaystyle{\int_{a}^{b}f=\int_{b}^{a}f}$;
$\displaystyle{\int_{a}^{b}f=-\int_{(a_{1},b_{2})}^{(b_{1},a_{2})}f}$;

  2. (b)

$\displaystyle{\int_{a}^{b}f=\int_{a}^{(c_{1},b_{2})}f+\int_{(c_{1},a_{2})}^{b}f}$;

  3. (c)

$\displaystyle{\int_{a}^{b}f=\int_{a}^{(b_{1},c_{2})}f+\int_{(a_{1},c_{2})}^{b}f}$;

  4. (d)

$\displaystyle{\int_{a}^{b}f=\int_{a}^{c}f+\int_{c}^{b}f+\int_{(a_{1},c_{2})}^{(c_{1},b_{2})}f+\int_{(c_{1},a_{2})}^{(b_{1},c_{2})}f}$.

###### Proof.

This follows immediately from Proposition 2. ∎

######  Proposition 33 (The double Newton integral is a primitive function).

Let $f$ be a double function defined on a double interval $I$. Suppose that
there exists a double primitive function $F$ of $f$ on $I$. Take $a\in I$ and
define the map $G:I\to\mathbb{R}$ by $G(x)\mapsto\int_{a}^{x}f$ for $x\in I$.

  1. (a)

The identity $\displaystyle{\Delta_{b}^{x}(G)=\Delta_{b}^{x}(F)}$ holds for
all $b,x\in I$.

  2. (b)

The function $G$ is double differentiable on $I$ and
$\displaystyle{G^{\prime}(b)=f(b)}$ for $b\in I$.

###### Proof.

Take $a,b,x\in I$. First we show (a). By Proposition 32 we get that:

$\begin{array}[]{rcl}\Delta_{b}^{x}(G)&=&\displaystyle{\int_{a}^{x}f+\int_{a}^{b}f-\int_{a}^{(x_{1},b_{2})}f-\int_{a}^{(b_{1},x_{2})}f}\\\\[10.0pt]
&=&\displaystyle{\int_{a}^{x}f+\int_{b}^{a}f+\int_{(a_{1},b_{2})}^{(x_{1},a_{2})}f+\int_{(a_{1},x_{2})}^{(b_{1},a_{2})}f}\
=\ \displaystyle{\int_{b}^{x}f\ =\ \Delta_{b}^{x}(F)}.\end{array}$

Next we show (b). By (a) we get that:

$\begin{array}[]{rcl}G^{\prime}(b)&=&\displaystyle{\lim_{x\leadsto
b}m_{b}^{x}(G)}\ =\ \displaystyle{\lim_{x\leadsto
b}\frac{\Delta_{b}^{x}(G)}{(x_{1}-b_{1})(x_{2}-b_{2})}}\\\\[10.0pt]
&=&\displaystyle{\lim_{x\leadsto
b}\frac{\Delta_{b}^{x}(F)}{(x_{1}-b_{1})(x_{2}-b_{2})}}\ =\
\displaystyle{\lim_{x\leadsto b}m_{b}^{x}(F)\ =\ F^{\prime}(b)\ =\
f(b).}\end{array}$

Alternatively, the equality $G^{\prime}=f$ follows from the fact that $G$
equals $F$ plus the double constant function $I\ni(x_{1},x_{2})\mapsto
F(a_{1},a_{2})-F(x_{1},a_{2})-F(a_{1},x_{2})$ which by Proposition 1 has zero
double derivative. ∎

######  Proposition 34 (The mean value theorem for double Newton integrals).

Let $f$ be a double function defined on a double interval $[a,b]$ for some
$a,b\in\mathbb{R}^{2}$ with $a<b$. Suppose that there exists a double
primitive function $F$ of $f$ on $[a,b]$. Then there exists $c\in(a,b)$ such
that:

$\int_{a}^{b}f=f(c)(b_{1}-a_{1})(b_{2}-a_{2}).$

###### Proof.

This follows from Propositions 14, 18 and 33. ∎

We now recall some classical notions (cf. e.g. [3]). Suppose that $f$ is a
double function defined on a double interval $[a,b]$ for some
$a,b\in\mathbb{R}^{2}$ with $a<b$. A _partition_ $P$ of $[a,b]$ is a choice of
points $x_{1,0},x_{1,1},\ldots,x_{1,m}\in[a_{1},b_{1}]$ and
$x_{2,0},x_{2,1},\ldots,x_{2,n}\in[a_{2},b_{2}]$ such that
$a_{1}=x_{1,0}<x_{1,1}<\cdots<x_{1,m-1}<x_{1,m}=b_{1}$ and
$a_{2}=x_{2,0}<x_{2,1}<\cdots<x_{2,n-1}<x_{2,n}=b_{2}.$ Given $P$, define the
$mn$ rectangles $R_{ij}=[x_{1,i-1},x_{1,i}]\times[x_{2,j-1},x_{2,j}]$, for
$1\leq i\leq m$ and $1\leq j\leq n$. The _norm_ $|P|$ of $P$ is the largest of
the diagonals in these $mn$ rectangles. Pick an arbitrary point
$(x_{1,i,j}^{*},x_{2,i,j}^{*})$ in each of the rectangles $R_{ij}$. For all
$i$ and $j$ put $\Delta x_{1,i}=x_{1,i}-x_{1,i-1}$ and $\Delta
x_{2,j}=x_{2,j}-x_{2,j-1}$. The corresponding _double Riemann sum_ is defined
as $R(f,P):=\sum_{i=1}^{m}\sum_{j=1}^{n}f(x_{1,i,j}^{*},x_{2,i,j}^{*})\Delta
x_{1,i}\Delta x_{2,j}.$ The double function $f$ is said to be _Riemann
integrable_ over $[a,b]$ and have _double integral_ $I=\iint_{[a,b]}f(x)\
dx_{1}dx_{2},$ if for every $\epsilon\in\mathbb{R}_{+}$ there is
$\delta\in\mathbb{R}_{+}$ such that $|R(f,P)-I|<\epsilon$ holds for every
partition $P$ of $[a,b]$ satisfying $|P|<\delta$ and for all choices of
$(x_{1,i,j}^{*},x_{2,i,j}^{*})$ in the subrectangles of $P$. If $f$ is
continuous on $[a,b]$, then $f$ is Riemann integrable over $[a,b]$ (see e.g.
[3, p. 293]).

######  Proposition 35 (The mean value theorem for double Riemann integrals).

Let $a,b\in\mathbb{R}^{2}$ and $a<b$. Suppose that $f:[a,b]\to\mathbb{R}$ is a
double function which is continuous. Then there exists $c\in(a,b)$ such that:

$\iint_{[a,b]}f(x)\ dx_{1}dx_{2}=f(c)(b_{1}-a_{1})(b_{2}-a_{2}).$

###### Proof.

See e.g. [3, p. 292]. ∎

######  Proposition 36 (The first double fundamental theorem of calculus).

Let $a,b\in\mathbb{R}^{2}$ and $a<b$. Suppose that $f:[a,b]\to\mathbb{R}$ is a
double function which is continuous. Define $G:[a,b]\to\mathbb{R}$ by
$G(x)=\iint_{[a,x]}f(x)\ dx_{1}dx_{2}$ for $x\in[a,b]$. Then $G$ is double
differentiable on $[a,b]$ with $G^{\prime}=f$.

###### Proof.

Take $x\in[a,b]$. We consider four cases.

Case 1: $x<b$. Take $h\in\mathbb{R}_{++}^{2}$ such that $x+h<b$. By
Proposition 35:

$m_{x}^{x+h}(G)=\frac{\iint_{[x,x+h]}f(x)\ dx_{1}dx_{2}}{h_{1}h_{2}}=f(t)$

for some $t\in(x,x+h)$. Letting $h\to 0^{++}$ yields that
$G^{\prime}_{++}(x)=f(x)$.

Case 2: $a<x$. Take $h\in\mathbb{R}_{--}^{2}$ such that $a<x+h$. By
Proposition 35:

$m_{x}^{x+h}(G)=\frac{\iint_{[x+h,x]}f(x)\
dx_{1}dx_{2}}{(-h_{1})(-h_{2})}=f(t)$

for some $t\in(x+h,x)$. Letting $h\to 0^{++}$ yields that
$G^{\prime}_{--}(x)=f(x)$.

Case 3: $(a_{1},x_{2})<(x_{1},b_{2})$. Take $h\in\mathbb{R}_{-+}^{2}$ such
that $x_{1}+h_{1}>a_{1}$ and $x_{2}+h_{2}<b_{2}$. By Proposition 35:

$m_{x}^{x+h}(G)=\frac{\iint_{[x_{1}+h_{1},x_{1}]\times[x_{2},x_{2}+h_{2}]}f(x)\
dx_{1}dx_{2}}{(-h_{1})h_{2}}=f(t)$

for some $t\in(x_{1}+h_{1},x_{1})\times(x_{2},x_{2}+h_{2})$. Letting $h\to
0^{-+}$ yields that $G^{\prime}_{-+}(x)=f(x)$.

Case 4: $(x_{1},a_{2})<(b_{1},x_{2})$. Take $h\in\mathbb{R}_{+-}^{2}$ such
that $x_{1}+h_{1}<b_{1}$ and $x_{2}+h_{2}>a_{2}$. By Proposition 35:

$m_{x}^{x+h}(G)=\frac{\iint_{[x_{1},x_{1}+h_{1}]\times[x_{2}+h_{2},x_{2}]}f(x)\
dx_{1}dx_{2}}{h_{1}(-h_{2})}=f(t)$

for some $t\in(x_{1},x_{1}+h_{1})\times(x_{2}+h_{2},x_{2})$. Letting $h\to
0^{+-}$ yields that $G^{\prime}_{+-}(x)=f(x)$.

Cases 1-4 show that $G$ is double differentiable on $[a,b]$ with
$G^{\prime}=f$. ∎

######  Proposition 37 (The second double fundamental theorem of calculus).

Let $f$ be a double function defined on a double interval $[a,b]$ for some
$a,b\in\mathbb{R}^{2}$ with $a<b$. Suppose that there exists a double
primitive function $F$ of $f$ on $[a,b]$. If $f$ is Riemann integrable on
$[a,b]$, then $\iint_{[a,b]}f(x)\ dx_{1}dx_{2}=\int_{a}^{b}f$.

###### Proof.

Take $\epsilon\in\mathbb{R}_{+}$ and put $I:=\int_{a}^{b}f$. Since $[a,b]$ is
a compact interval, $f$ is uniformly continuous on $[a,b]$ (see e.g. [8,
Theorem 4.19]). Therefore, there exists $\delta\in\mathbb{R}_{+}$ such that:

$|f(c)-f(d)|<\frac{\epsilon}{(b_{1}-a_{1})(b_{2}-a_{2})}$

whenever $c,d\in[a,b]$ and
$\sqrt{(c_{1}-d_{1})^{2}+(c_{2}-d_{2})^{2}}<\delta$. Consider a fixed double
Riemann sum:

$R:=\sum_{i=1}^{m}\sum_{j=1}^{n}f(x_{1,i,j}^{*},x_{2,i,j}^{*})\Delta
x_{1,i}\Delta x_{2,j}$

defined by a partition $P$, with $|P|<\delta$, and a choice of points
$(x_{1,i,j}^{*},x_{2,i,j}^{*})$ in the corresponding rectangles. We wish to
show that $|R-I|<\epsilon$. By Proposition 34 there exist
$c_{i,j}\in(x_{1,i-1},x_{1,i})\times(x_{2,j-1},x_{2,j})$ with:

$\int_{(x_{1,i-1},x_{2,j-1})}^{(x_{1,i},x_{2,j})}f=f(c_{i,j})\Delta
x_{1,i}\Delta x_{2,i}$

for $i=1,\dots,m$ and $j=1,\ldots,n$. By repeated application of Proposition
32(b)(c) we therefore get that:

$I=\sum_{i=1}^{m}\sum_{j=1}^{n}\int_{(x_{1,i-1},x_{2,j-1})}^{(x_{1,i},x_{2,j})}f=\sum_{i=1}^{m}\sum_{j=1}^{n}f(c_{i,j})\Delta
x_{1,i}\Delta x_{2,i}$

which in turn implies that:

$\begin{array}[]{rcl}\displaystyle{\left|R-I\right|}&=&\displaystyle{\left|\sum_{i=1}^{m}\sum_{j=1}^{m}(f(x_{1,i,j}^{*},x_{2,i,j}^{*})-f(c_{i,j}))\Delta
x_{1,i}\Delta x_{2,i}\right|}\\\\[20.0pt]
&\leq&\displaystyle{\sum_{i=1}^{m}\sum_{j=1}^{m}\left|f(x_{1,i,j}^{*},x_{2,i,j}^{*})-f(c_{i,j})\right|\Delta
x_{1,i}\Delta x_{2,i}}\\\\[15.0pt]
&<&\displaystyle{\frac{\epsilon}{(b_{1}-a_{1})(b_{2}-a_{2})}\sum_{i=1}^{m}\sum_{j=1}^{m}\Delta
x_{1,i}\Delta x_{2,i}}\\\\[15.0pt]
&=&\displaystyle{\frac{\epsilon}{(b_{1}-a_{1})(b_{2}-a_{2})}\cdot(b_{1}-a_{1})(b_{2}-a_{2})=\epsilon.}\\\\[10.0pt]
\end{array}$

We have now shown that $\iint_{[a,b]}f(x)\ dx_{1}dx_{2}=\int_{a}^{b}f$. ∎

Let $f$ be a double function defined on a double _open_ interval $(a,b)$ for
some $a,b\in\overline{\mathbb{R}}^{2}$ with $a<b$. Suppose that there exists a
double primitive function $F$ of $f$ on $(a,b)$. If the double signed limit:

(2) $\lim_{y\to b^{--};\ x\to a^{++}}\Delta_{x}^{y}(F)$

exists, then we say that $\int_{a}^{b}f$ is a _convergent_ improper double
Newton integral with _value_ equal to the limit (2). If the limit in (2) does
not exist, then we say that the improper double Newton integral
$\int_{a}^{b}f$ is _divergent_. Note that if all of the following four signed
limits exist:

$\begin{array}[]{lcl}\displaystyle{A:=\lim_{x\to
b^{--}}F(x)}&&\displaystyle{B:=\lim_{x\to a^{++}}F(x)}\\\\[15.0pt]
\displaystyle{C:=\lim_{x\to(b_{1},a_{2})^{-+}}F(x)}&&\displaystyle{D:=\lim_{x\to(a_{1},b_{2})^{+-}}F(x)}\end{array}$

then $\int_{a}^{b}f$ is convergent with value equal to $A+B-C-D$.

######  Example 38.

(a) Suppose that $F$ and $f$ are the double functions defined on
$I:=(0,1)\times(0,1)$ by $F(x)=(x_{1}+x_{2})\ln(x_{1}+x_{2})$ and
$f(x)=1/(x_{1}+x_{2})$ for $x\in I$. By Proposition 10, $F$ is double
differentiable on $I$ with $F^{\prime}=f$. Since
$\lim_{x\to(1,1)^{--}}F(x_{1},x_{2})=2\ln(2)$ and
$\lim_{x\to(0,0)^{++}}F(x_{1},x_{2})=\lim_{x\to(1,0)^{-+}}F(x_{1},x_{2})=\lim_{x\to(0,1)^{+-}}F(x_{1},x_{2})=0$
the improper double Newton integral $\int_{(0,0)}^{(1,1)}1/(x_{1}+x_{2})$ is
convergent with value equal to $2\ln(2)$.

(b) Suppose that $F$ and $f$ are the double functions defined on
$I:=(0,1]\times(0,1]$ by $F(x)=x_{1}/(x_{1}+x_{2})$ and
$f(x)=(x_{1}-x_{2})/(x_{1}+x_{2})^{3}$ for $x\in I$. By Proposition 10, $F$ is
double differentiable on $I$ with $F^{\prime}=f$. However, since $\lim_{s\to
0^{+}}\Delta_{(s,s)}^{(1,1)}(F)=0$ and $\lim_{t\to
0^{+}}\Delta_{(t,2t)}^{(1,1)}(F)=-1/6$ the improper double Newton integral
$\int_{(0,0)}^{(1,1)}(x_{1}-x_{2})/(x_{1}+x_{2})^{3}$ is divergent.

Many standard calculus textbook problems concerning double integrals over non-
rectangular regions are solved by iterated integration. If, however, the
region in question can be mapped bijectively onto a double interval, then such
integrals can instead be considered as improper double Newton integrals. In
fact, by Proposition 37 and the result in [9] we get the following:

######  Proposition 39.

Let $a,b\in\overline{\mathbb{R}}^{2}$ with $a<b$. Suppose that $D$ is an open
subset of $\mathbb{R}^{2}$ and that $h:(a,b)\to D$ is a bijection such that
$h$ and $h^{-1}$ are continuous and have continuous partial derivatives. Let
$J(x)$ denote the absolute value of the Jacobian determinant of $h$ at
$x\in(a,b)$. If $f:D\to\mathbb{R}$ is a function which is integrable on $D$
and $g:=(f\circ h)\cdot J$ has a double primitive function on $(a,b)$, then
the improper double Newton integral of $g$ from $a$ to $b$ is convergent and
$\iint_{D}f(x)\ dx_{1}dx_{2}=\int_{a}^{b}g$.

######  Example 40.

(a) We wish to evaluate $\iint_{D}f(x)\ dx_{1}dx_{2}$ where $f(x)=x_{1}x_{2}$
for $x\in D$, and $D$ is the interior of the triangle with vertices $(0,0)$,
$(1,0)$ and $(1,1)$. Define $h:(0,1)\times(0,1)\to D$ by $h(u,v)=(u,uv)$. Then
$J(u,v)=u$ and $\iint_{D}f(x)\ dx_{1}dx_{2}=\int_{(0,0)}^{(1,1)}u\cdot uv\cdot
u=\int_{(0,0)}^{(1,1)}u^{3}v=\Delta_{(0,0)}^{(1,1)}(u^{4}v^{2}/8)=1/8$.

(b) We wish to evaluate $\iint f(x)\ dx_{1}dx_{2}$ where
$f(x)=1/(x_{1}+x_{2})^{2}$ for $x\in D$, and $D$ is defined by $0<x_{1}<1$ and
$0<x_{2}<x_{1}^{2}$. Define $h:(0,1)\times(0,1)\to D$ by $h(u,v)=(u,u^{2}v)$
for $u,v\in(0,1)$. Then $J(u,v)=u^{2}$ and $\iint_{D}f(x)\
dx_{1}dx_{2}=\int_{(0,0)}^{(1,1)}u^{2}/(u+vu^{2})^{2}=\int_{(0,0)}^{(1,1)}1/(1+uv)^{2}=$
$\Delta_{(0,0)}^{(1,1)}(\ln(1+uv))=\ln(2)$.

(c) We wish to evaluate $\iint f(x)\ dx_{1}dx_{2}$ where $f(x)=e^{-x_{1}^{2}}$
for $x\in D$, and $D$ is defined by $x_{1}>0$ and $-x_{1}<x_{2}<x_{1}$. Define
$h:(0,\infty)\times(-1,1)\to D$ by $h(u,v)=(u,uv)$ for $u\in(0,\infty)$ and
$v\in(-1,1)$. Then $J(u,v)=u$ and $\iint_{D}f(x)\
dx_{1}dx_{2}=\int_{(0,-1)}^{(\infty,1)}e^{-u^{2}}u=\lim_{s\to\infty}\Delta_{(0,-1)}^{(s,1)}(-e^{-u^{2}}v/2)=1$.

(d) We wish to evaluate $\iint f(x)\ dx_{1}dx_{2}$ where
$f(x)=1/(x_{1}+x_{2})$ for $x\in D$, and $D$ is defined by $1<x_{1}$ and
$0<x_{2}<1/x_{1}$. Define $h:(1,\infty)\times(0,1)\to D$ by $h(u,v)=(u,v/u)$
for $u\in(1,\infty)$ and $v\in(0,1)$. Then $J(u,v)=u^{-1}$ and $\iint_{D}f(x)\
dx_{1}dx_{2}=\int_{(1,0)}^{(\infty,1)}(u+v/u)^{-1}\cdot
u^{-1}=\int_{(1,0)}^{(\infty,1)}(u^{2}+v)^{-1}=\lim_{t\to\infty;\ s\to
0^{+}}\Delta_{(1,s)}^{(t,1)}\left(u\ln(u^{2}+v)+2\sqrt{v}\arctan(u/\sqrt{v})\right)=\pi/2-\ln(2)$.

##  6\. Triple calculus, quadruple calculus and beyond

It is easy to work out the corresponding difference operators in higher
dimensions. Namely, if $a,b\in\mathbb{R}^{n}$ and $f$ is an $n$-variable
function, then:

$\Delta_{a}^{b}(f)\ \ =\
\sum_{s\in\\{0,1\\}^{n}}(-1)^{s}f(s_{1}a_{1}+(1-s_{1})b_{1},\ldots,s_{n}a_{n}+(1-s_{n})b_{n})$

where $(-1)^{s}:=(-1)^{s_{1}+s_{2}+\cdots+s_{n}}$. So, for instance, if $n=3$,
then we get that:

$\begin{array}[]{rcl}\Delta_{a}^{b}(f)&=&f(b_{1},b_{2},b_{3})\ -\
f(a_{1},b_{2},b_{3})\ -\ f(b_{1},a_{2},b_{3})\ -\
f(b_{1},b_{2},a_{3})\\\\[5.0pt] &+&f(a_{1},a_{2},b_{3})\ +\
f(a_{1},b_{2},a_{3})\ +\ f(b_{1},a_{2},a_{3})\ -\
f(a_{1},a_{2},a_{3}).\end{array}$

In [1, 2] higher-dimensional analogues of all the results established in this
article are shown to hold, that is there is also a triple calculus, a
quadruple calculus and beyond, at our disposal.

It seems to the author of the present article that Bögel did not consider
higher-dimensional versions of Schwarz’s theorem. To be more precise, suppose,
for instance, that $f$ is a five-variable function, that $f$ is double
differentiable with respect to the first two variables, with double derivative
denoted by $f^{\prime}_{12}$, and that $f$ is triple differentiable with
respect to the last three variables, with triple derivative denoted by
$f^{\prime}_{345}$. If we also suppose that the iterated derivatives
$(f^{\prime}_{345})^{\prime}_{12}$ and $(f^{\prime}_{12})^{\prime}_{345}$
exist and are double respectively triple continuous, does it then follow that
$f$ is quintuple differentiable with
$f^{\prime}_{12345}=(f^{\prime}_{345})^{\prime}_{12}=(f^{\prime}_{12})^{\prime}_{345}$?
Since the proof of Proposition 10 only depends on the mean value theorem in
each variable, it seems reasonable to believe that this proof is generalizable
to higher dimensions if we use Bögel’s higher-dimensional mean value theorem.

Another classical result from calculus that neither we nor Bögel has
considered is Darboux’s theorem. Recall that this result states that if a
single variable function is differentiable on an open interval, then the
derivative enjoys the intermediate value property on this interval. It is not
clear to the author of the present article if there is a double (or higher-
dimensional) analogue of this result. Note that the usual text book proof for
Darboux’s theorem uses the Weierstrass extreme value theorem (see e.g. [6]),
which, as we have pointed out earlier, is not at our disposal for double
functions. However, there are proofs of Darboux’s theorem which are based only
on the mean value theorem for differentiable functions and the intermediate
value theorem for continuous functions (see loc. cit.). Therefore, it is
plausible that there indeed is a double (and higher) version(s) of Darboux’s
theorem which is (are) reachable by the methods used in this article.

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