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The function $f\left(x\,y\right)$ is differentiable if it has a differential.
The differential of $f\left(x\,y\right)$ exists if $\mathrm{\Δ}fequals;f\left(xplus;hcomma;yplus;k\right)f\left(xcomma;y\right)$ can be written as
$A\left(x\,y\right)hplus;B\left(xcomma;y\right)kplus;\mathrm{eta;}\left(hcomma;k\right)\sqrt{{h}^{2}plus;{k}^{2}}$
where $\mathrm{\η}\left(h\,k\right)\to 0$ as $\left(h\,k\right)\to \left(0\,0\right)$.
If $\mathrm{\Δ}f$ has such a representation, then it necessarily follows that $A\={f}_{x}$ and $B\={f}_{y}$ and the expression $\mathrm{df}\={f}_{x}\left(x\,y\right)\mathrm{dx}plus;{f}_{y}\left(xcomma;y\right)\mathrm{dy}$ is called the exact, or total differential of $f$.
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Table 4.11.1 lists three functions that serve as counterexamples for clarifying the discussion of differentiability and its connection to partial derivatives and continuity.
Function

Properties

$f\left(x\,y\right)\=\{\begin{array}{cc}\left({x}^{2}\+{y}^{2}\right)\mathrm{sin}\left(\frac{1}{\sqrt{{x}^{2}\+{y}^{2}}}\right)& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=0\,0\)\end{array}$

•

First partials exist and are bounded but are not continuous


$g\left(x\,y\right)\=\{\begin{array}{cc}\left({x}^{2}\+{y}^{2}\right)\mathrm{sin}\left(\frac{1}{{x}^{2}\+{y}^{2}}\right)& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

•

First partials exist, are not bounded and are not continuous


$h\left(x\,y\right)\=\{\begin{array}{cc}\frac{xy\left({x}^{2}{y}^{2}\right)}{{x}^{2}\+{y}^{2}}& \left(x\,y\right)\ne \left(0\,0\right)\\ 0& \left(x\,y\right)\=\left(0\,0\right)\end{array}$

•

First partials exist and are continuous

•

Second partials exist but are not continuous

•

Hence, ${h}_{\mathrm{xy}}\left(0\,0\right)\ne {h}_{\mathrm{yx}}\left(0\,0\right)$

•

In class ${C}_{1}$ but not ${C}_{2}$


Table 4.11.1 Three counterexamples



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The Venn diagram in Figure 4.11.1 helps clarify the contrasting properties of the three functions in Table 4.11.1.

Figure 4.11.1 Venn diagram for the functions and properties in Table 4.11.1



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The functions $f$ and $g$ are differentiable, but their first partials are not continuous, so they are not in continuity class ${C}_{1}$. The function $f$ is differentiable, and its first partials are continuous, so it is in continuity class ${C}_{1}$, but not ${C}_{2}$ because its second partials, which exist, are not continuous. Moreover, the mixed partials for this function are not equal. If ${C}_{0}$ represents the set of continuous functions; and D, the set of differentiable ones, then Figure 4.11.1 suggests the following set inclusions: ${C}_{0}\supset \mathrm{D}\supset {C}_{1}\supset {C}_{2}$.
Table 4.11.2 states three theorems relating differentiability, continuity, and partial derivatives.
A differentiable function is continuous.

A sufficient (but not necessary) condition for differentiability is the continuity of the first partial derivatives.

A sufficient (but not necessary) condition for the equality of the mixed second partials is their continuity.

Table 4.11.2 Three theorems relating differentiability, continuity, and partial derivatives


