Show that the function $$f(x, y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{xy}{x^{2}+y^{2}} & \hbox{if} & (x,y)\neq (0,0) \\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$$

has partial derivatives at $(0,0),$ but is not continuous at that point.

836

Solution We use the definition to find the partial derivatives of $f$ at $(0,0)$.

The partial derivatives $f_{x}(0,0)$ and $f_{y}(0,0)$ both exist.

But $\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{xy}{x^{2}+y^{2}}$ does not exist. (See Example 2 from Section 12.2.) As a result, $f$ is not continuous at $(0,0)$.