TITLE: Why is continuous differentiability important?
QUESTION [9 upvotes]: In calculus, I would presume that the notion of continuous differentiability is important, which is why we have classes $C^1, C^2,\ldots,C^n$ which are defined in terms of having a continuous $n$th derivative. But why? Why is the derivative being continuous relevant at all?

What is the motivation for defining $C^n$ in terms of not merely being $n$ times differentiable, but $n$ times continuously differentiable? For which (important) theorems in single and multivariable calculus is the hypothesis of continuous differentiability absolutely required? 

It is not required for either the fundamental theorem of calculus or integration by substitution, though it is often presented as being such.

REPLY [4 votes]: One reason $C^1$ is important is its practicality. Namely, there is a theorem that if $f$ is $C^1$ on an open set $U$ then $f$ is differentiable at all points of $U$. It's usually pretty easy to check $C^1$: often you simply look at the form of the coordinate functions of $C^1$ and observe, from your knowledge of elementary calculus, that they are differentiable and their derivatives are continuous. And once you've completed that check, voila, you conclude that $f$ is differentiable.
$C^2$ is also practical, namely via the theorem on equality of mixed 2nd partials (and, similarly, $C^n$ implies equality of mixed $n$th partials).