Show that we can reorder mixed partials, if every partial is continuous

Question

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then $$D_{i_1i_2...i_n}f = D_{i{\sigma (1)}i{\sigma (2)}...i{\sigma (n)}}f$$ Basically, prove that we can reorder mixed partials however we like, as long as every partial is continuous.

Answer 1

Each permutation $\sigma$ can be presented as a concatenation of transpositions, i.e. you can reach $D_{i{\sigma (1)}i{\sigma (2)}...i{\sigma (n)}}f$ by subsequently change two partial derivatives. You can apply Schwarz' theorem to prove, that you can change two partial derivatives.