Condition for equality of mixed derivatives 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.

I think that the confusion is in perhaps too much detail. Theorem 12.11 states the "existence" of derivatives in c and the continuity of other of other derivatives in n-ball, is a sufficient condition, for making f differentiable.
Theorem 12.12 states, the existence of both partial derivatives and and both are differentiable at c, then the relation is exact, and commutable. (Very interesting result that gets used extensively in Electromagnetic/Quantum Fields... but that's the digression.)
Theorem 12.13, which adds "continuity" in Theorem 12.12, flows straight from the 12.12, in my view, there never was the separate need for 12.13, as the "continuity" could've been included/stated in the 12.12, clearing up the verbiage a bit, perhaps.
And the stated conditions are "sufficient" to determine the "uniqueness", please read the condition in their geometric perspective to ascertain the uniqueness of the conditions, implied as to sufficiently adjudicate the behaviour of f, hence their ability to ascertain the differentiability and continuity, and the "sufficient-ness" of this ability.
