Usually in most introductory Calculus courses, a definition of differentiability at a point $a$ is defined, as follows :
A function $f$ is differentiable at $a$ if $f'(a)$ exists
As a corollary of this :
A function $f$ is not differentiable at $a$ if $f'(a)$ does not exist
However can differentiability be defined another way, in terms of continuity of the derivative function?
Possible Alternate Definition
A function $f$ is differentiable at $a$ if $f'(x)$ is continuous at $a$
Stated more formally, given $f : \mathbb{R} \to \mathbb{R}$, $f$ is differentiable $\text{iff}$
$$\lim_{x\to a} f'(x) = f'(a)$$
A simple example to illustrate this would be to look at $f(x) = |x|$.
$$f'(x) =\begin{cases} \ \ \ 1 &\text{if} & x > 0 \\ -1 &\text{if} & x < 0 \\ \end{cases}$$
And using the alternate definition, we can show that since the limit doesn't exist $$\lim_{x \to 0} f'(x) \not= f'(0)$$ and thus $f'(x)$ is discontinous at $x=0$, and hence not differentiable at $x=0$.
Note: I haven't looked up whether this is actually a definition or not (this is just something I thought up), as far as I know, differentiability in most textbooks are not defined in this way. If this possible alternate definition, that I've given is incorrect, in any way whatsoever, or if it doesn't generalize well (conceptually that is) outside of single-variable calculus for real-valued functions, please feel free to tear it apart completely.