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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 $\partial f^n/\partial x^n$, $n>k$, n>>k$, exists and continuous. Can one say anything about $f_x$ better than $C^{k-1}$?
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Higher order partial derivatives and global regularity.

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. 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 $C^{k-1}$?