Function

Properties

$f\left(x\,y\right)\=\{\begin{array}{cc}\left(x^2\+y^2\right)\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}$

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

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

Table 4.11.1   Three counterexamples

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

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

Example 4.11.1

Show that the function $f\left(x\,y\right)$ in Table 4.11.1 has first partial derivatives everywhere.

Example 4.11.2

Show that the function $f\left(x\,y\right)$ in Table 4.11.1 has a differential at the origin, and hence, is differentiable at the origin.

Example 4.11.3

Show that the first partial derivatives of the function $f\left(x\,y\right)$ in Table 4.11.1 are not continuous even though they are bounded.

Example 4.11.4

Show that the function $g\left(x\,y\right)$ in Table 4.11.1 has first partial derivatives everywhere.$$

Example 4.11.5

Show that the function $g\left(x\,y\right)$ in Table 4.11.1 has a differential at the origin, and hence, is differentiable at the origin.

Example 4.11.6

Show that the first partial derivatives of the function $g\left(x\,y\right)$ in Table 4.11.1 are not bounded, and are not continuous.

Example 4.11.7

Find the first partial derivatives for the function $h\left(x\,y\right)$ in Table 4.11.1.

Example 4.11.8

Show that the first partial derivatives of the function $h\left(x\,y\right)$ in Table 4.11.1 are continuous.

Example 4.11.9

Show that the function $h\left(x\,y\right)$ in Table 4.11.1 has a differential at the origin, and hence is differentiable at the origin.

Example 4.11.10

Find the second partial derivatives for the function $h\left(x\,y\right)$ in Table 4.11.1. As a consequence of this construction, establish that $h_{\mathrm{xy}}\left(0\,0\right)\ne h_{\mathrm{yx}}\left(0\,0\right)$.