TITLE: Condition for equality of mixed derivatives
QUESTION [6 upvotes]: It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of $D_{r,r}f$ and $D_{k,k}f$? Here $f$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $D_kf$ denotes the partial derivative of $f$ w.r.t. the $k^{th}$ variable.



The images are from Tom Apostol's Mathematical Analysis.

REPLY [0 votes]: The condition that $D_{r,k} f, D_{k,r}$ are continuous in an n-ball $B(c)$ implies that $D_{r,r}, D_{k,k}$ exit.
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