Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.

- Is it true that $f_{xy}$ exists and continuous?
- Is it true that $f_{yx}$ exists and continuous?

I suspect that the answers are negative, so let me ask a more general question.

**Question**
*
If f is $C^k$ and $\partial f^n/\partial x^n$, $n>>k$, exists and continuous. Can one say anything about $f_x$ better than $C^{k-1}$?*