What is the importance of functions of class $C^k$? 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?
 A: One reason is that it's desirable to equip a class of functions with a norm, particularly with a norm which makes the class a complete metric space, i.e., Banach space. In $C^k$ we can use the supremum of the $k$th derivative (plus lower-order terms to make the norm nonzero on polynomials of degree less than k). A general differentiable function may well have  an unbounded derivative, e.g., $f(x)=x^2\sin e^{1/x}$. We could try to look the space of functions with bounded $k$th derivative, denoting it $B^k$. I can't tell off the top of my head if $B^k$ is complete, but it loses to $C^k$ in another important aspect: polynomials are dense in $C^k$ but not in $B^k$. 
Conclusion: if you want a complete normed space in which the norm has to do with the supremum of the $k$th derivative, and in which polynomials are dense, then $C^k$ is what you have to use. 
If you decide to use the integral of the $k$th derivative instead, you get Sobolev spaces. 
A: 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.
A: Well, first of all, if the derivative of order $k$-$1$ is not continuous, then the $k$-th derivative does not even exist.
As for the last one (the $k$-th one), you have a point: why do we need that derivative to be continuous. If we drop this requirement, wouldn't we get a broader class of functions, maybe able to represent a broader class of phenomena? In some sense yes. But...
While working with $C^k$ spaces, we are often interested in the pointwise value of some function $f$. If the 3rd derivative of $f$ has a physical/sociologic/demographic/whatever meaning, then it would be probably not satisfying if that derivative jumped at a point $x_0$. I said probably, since there are phenomena involving quantities that are allowed to change suddenly. 
In physics, for instance, the electric charge density can vary discontinuously. This quantity, in dimension one, turns out to be the derivative of the electric field, which itself is the derivative of the electric potential. Therefore, for this phenomenon, a potential which is $C^2$ according to the standard definition of $C^k$ spaces, might not represent the phenomenon that I am observing, since it would give a continuous charge density.
Nevertheless, functions that have $k$ continuous derivatives and the ($k$+$1$)-th has only jump discontinuity, form a pretty interesting space, called Hölder space with exponent $1$ (written $C^{k,1}$). Hölder spaces are pretty complicate, in my opinion, but the case $k=0$ is fairly easy: corresponds to Lipschitz-continuous functions, which are kind of useful in some contest. Maybe these spaces are closer to what you would expect.
But as I said, in many applications you are interested in the pointwise value of a function, and you don't expect that function to jump.
Things become more interesting when you don't care what's the value of the function at any given point, but rather you care that some other property holds, like for instance 
$$\int_a^b |f(x)|^2dx<\infty$$
But this is completely another story...
