Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be continuously differntiable, with continuous $f_{xy}$. If $\exists f_{yx}$, show $f_{xy} = f_{yx}$.
I see that this is weaker than simply stating that $f \in C''$, since a well known theorem shows that $f_{xy} = f_{yx}$. I need help with using the $f_{xy}$ continuous condition to prove the statement.
Also, can do I generalize this statement for $f: \mathbb{R}^n \rightarrow \mathbb{R}$? How many continuity statements would I need?