Recently I was trying to construct a counterexample to the statement “If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are continuous at $(0,0)$, then $f$ is twice differentiable at $(0,0)$“. In order to do that the following question arose:

Is there a function $fcolon mathbb{R}^2tomathbb{R}$ with bounded $f_x$, $f_y$, $f_{xx}$, $f_{yy}$, which is non-differentiable at $(0,0)$?

If there exists such a function, then we’re done. The functions which came to mind have unbounded second partials $f_{xx}$ and $f_{yy}$.