I'm confused about this theorem which is sometimes associated to the Darboux theorem for real functions.
Let $f: dom(f)\subseteq\mathbb{R} \rightarrow \mathbb{R}$ be a function continuous in a point $x_0 \in dom(f)$ and differentiable in a neighbourhood of $x_0$ (except for the point $x_0$).
If the $\lim_{x\to x_0}f'(x)=l \in \mathbb{R} $ then $f'(x_0)=\lim_{x\to x_0}f'(x)$.
The theorem is easily proved using de l'Hopital theorem.
I don't really understand the consequences and the meaning of this theorem.
If a function is such that $f(x_0)=\lim_{x\to x_0}f(x)$ the function $f$ is continuous in $x_0$ , so does the theorem imply that, under those conditions, the derivative of the function is continuous in $x_0$ ?
I read somewhere that this theorem (and the one of Darboux) imply that $f'$ does not have jump discontinuities, but nothing else can be said about it. If so, I can't understand how $f'$ could have other kinds of discontiuities, removable or essential.
What can one say about the continuity of $f'$, using this theorem (and the theorem of Darboux)?
Thanks in advice for your help
