Clairaut Theorem Counterexample

Question

Do you know one function $f:\mathbb{R}^2\to\mathbb{R}$ such that $f_{xy}(a,b)=f_{yx}(a,b)$ at point $(a,b)$ but $f_{xy}$ are not continuous at $(a,b)$?

Answer 1

Consider the function:

$f\left( x,y \right)=\left\{ \begin{align} & \frac{{{x}^{2}}{{y}^{2}}}{{{x}^{2}}+{{y}^{2}}},\quad \left( x,y \right)\ne \left( 0,0 \right) \\ & 0\quad \quad \quad ,\quad \left( x,y \right)\ne \left( 0,0 \right) \\ \end{align} \right.$

It can be shown that ${{f}_{xy}}\left( 0,0 \right)={{f}_{yx}}\left( 0,0 \right)$ but $\underset{\left( x,y \right)\to \left( 0,0 \right)}{\mathop{\lim }}\,{{f}_{yx}}\left( x,y \right)\ne 0={{f}_{yx}}\left( 0,0 \right)$