It is all known that there are some functions whose mixed second parital derivatives exist but not equal at a single point, I am thinking what is about on a connected set.
To be clear, Is there any function $f$ defined on an open connected set $D\subset \mathbf{R}^2$ such that
- For all $(x,y)\in D$, $$\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial ^2f}{\partial x\partial y},\frac{\partial^2 f}{\partial y\partial x}$$ exist.
- For all $(x,y)\in D$, $$ \frac{\partial ^2f}{\partial x\partial y}\neq \frac{\partial^2 f}{\partial y\partial x}.$$