I had a little difficulty understanding Theorem 12.13 of Apostol's Mathematical Analysis, and was wondering if someone who had read this book could give me some pointers. Specifically, on page 360, the author writes
"As a consequence of Theorem 12.11 and 12.12 we have
Theorem 12.13 If both partial derivatives $D_{r}f$ and $D_{k}f$ exist in an n-ball $B(c)$ and if both $D_{k,r}f$ and $D_{r,k}f$ are continuous at c, then $D_{k,r}f(c) = D_{r,k}f(c)$."
My understanding is that we're using Theorem 12.11 to establish the differentiability of $D_{r}f$ and $D_{k}f$ so that we can use Theorem 12.12 to assert the equality of $D_{k,r}f(c) = D_{r,k}f(c)$. If so, my question is: why shouldn't we require $D_{r,r}f$ and $D_{k,k}f$ also exist at c?
For convenience, I typed Theorem 12.11 & 12.12 below:
"Theorem 12.11 Assume that one of the partial derivatives $D_{1}f,...,D_{n}f$ exists at c and that the remaining n-1 partial derivatives exist in some n-ball $B(c)$ and are continuous at c. Then $f$ is differentiable at c.
Theorem 12.12 If both partial derivatives $D_{r}f$ and $D_{k}f$ exist in an n-ball $B(c)$ and if both are differentiable at c, then $D_{k,r}f(c) = D_{r,k}f(c)$."
