TITLE: What is the importance of functions of class $C^k$?
QUESTION [9 upvotes]: In all calculus textbooks, after the part about successive derivatives, the $C^k$ class of functions is defined.
The definition says : 

A function is of class $C^k$ if it is differentiable $k$ times and the $k$-th derivative is continuous.

Wouldn't be more natural to define them to be the class of functions that are differentiable $k$ times?
Why is the continuity of the $k$th derivative is so important so as to justify a specific definition?

REPLY [4 votes]: You can certainly consider $k$-times differentiable functions on, say, $[a,b]\subset \mathbb R$ and  give them a notation,  like $D^k[a,b]$.
The point however is that many well-known and interesting theorems, true for $C^k[a,b]$, will fail for $D^k[a,b]$or won't even make sense. Here are three examples from elementary calculus:      
a) Integration by parts for $u,v\in C^1[a,b]$: $$\int_{a}^{b}u(x)v^{\prime }(x)dx=\left(
u({b})v(b)-u(a)v(a)\right) -\int_{a}^{b}u^{\prime
}(x)v(x)dx$$ The integrals don't even make sense a priori if $u', v'$ are not continuous.  
b) Taylor's formula 
$$f(x)=\sum_{i=0}^k\frac{f^{(i)}(a)}{i!}(x-a)^i+\int_a^x\frac{f^{(k+1)}(t)}{k!}(x-t)^kdt\quad (x\in [a,b])$$ is valid for $f\in C^{k+1}([a,b])$ and again doesn't make sense for $f\in D^{k+1}([a,b])$    
c) The change of variables formula $$  \int_a^b f(\phi (t))\phi'(t)dt=\int_{\phi(a)}^{\phi(b)}  f(x)dx            $$ again necessitates that $\phi$ be a $C^1$-function and not merely a differentiable one.