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?