Here is a well-known theorem of Analysis:
Let $f$ be a real-valued function that is continuous on an open interval $I$ of $\mathbb R$ and $a$ a point of $I$. We assume that $f$ is differentiable on $I\setminus\{a\}$. Then, if $\lim_{x\to a}f'(x)=L$, then $f$ is differentiable at $a$ and $f'(a)=L$.
See here for the proof (of the slightly different statement).
I discovered that theorem recently, when a student used it in examination.
What are some good applications of the quoted theorem?
I am looking for examples from the theory, as well as concrete applications. For example, a case where $f'(a)$ is difficult to compute from the usual Rules/the definition, but $f'(x)$ is easy to compute for $x\neq a$, and $\lim_{x\to a}f'(x)$ is also easy to determine. But, I failed to construct an example.
Any idea?
