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


1. Is it true that $f\_{xy}$ exists and continuous?
2. 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 $C^{k-1}$?*



Higher order partial derivatives and global regularity.
From Counterexamples in Analysis (9.10), consider the function $f(x) = xy\frac{x^2-y^2}{x^2 +y^2}$. A bit of calculation confirms that $f\_{xx}$ exists and is continuous, while $f\_{xy}$ is discontinuous at the origin. 


In general, consider, for example, 
$f(x,y) = \frac{x^k y^3}{x^2+y^2}$. 
You can take any number of $x$ derivatives, and the result will still be continuous, but if you take $k$ $x$-derivatives, and then a $y$-derivative, you will have a term of the form $\frac{y^2}{x^2+y^2}$ in your sum, which is problematic at the origin. So you can't get better than $C^k$ for this function. 


