Symmetry of second (and higher) order partial derivatives

We've learned today that if $f$ has second order partial derivatives that are continuous at some point $a$, then they're all equal to each other at that point.

Then there's a short remark that says this holds for higher order as well, i.e. $$D_1D_2D_3f= D_3D_2D_1f$$

What's the proof of this? I mean generally, and not just in above example?

Also, what kind of function wouldn't satisfy these conditions? I've been trying to think of a simple function that wasn't $C^2$, even if just in some point, and it's proving to be more difficult than I'd thought. Does such a simple function exist, without entering the realm of complicated piecewise functions?

For the proof of why it holds for higher order derivatives, remember that the derivatives are themselves functions so \begin{align}D_1D_2D_3f &= D_1D_2(D_3f) = D_2D_1(D_3 f) \\&= D_1(D_2 D_3 f) = D_1 (D_2 D_3 f)\end{align} by the fact we can interchange the order for the second derivative of a function. In general applying the above gives us the permutations $D_iD_jD_k = D_jD_iD_k$ and $D_iD_jD_k = D_iD_kD_j$. Using these two permutations repeatedly we can arrive at any order of $D_1$, $D_2$ and $D_3$.