A continuous $f$ defined on $[c,b)$ is differentiable on $(c,b)$ and if $\lim_{x\to c+0} f'(x)=+\infty$, $f$ is not differentiable at $c$. I asked the following question and Paul Frost answered my question.
A continuous $f$ defined on $[c,b)$ is differentiable on $(c,b)$ and if $\lim_{x\to c+0} f'(x)$ doesn't exist, $f$ is not differentiable at $c$.

I want to ask a similar question:
Does the following proposition hold or not?

Proposition 2:
If a continuous function $f(x)$ defined on $[c,b)$ is differentiable on $(c,b)$ and if $\lim_{x\to c+0} f'(x)=+\infty$ or $\lim_{x\to c+0} f'(x)=-\infty$, then $f(x)$ is not differentiable at $c$.

 A: Advice for questions about derivatives, i.e., functions that are everywhere in some interval the derivative of some other function.
These are elusive little devils.  As a calculus student you expect them to be continuous since you never see examples that are not continuous.  Then as an analysis student you see examples with a few discontinuities and wonder just how many discontinuities a derivative might have.  [A number of StackExchange questions on that topic.]
But the very first step in understanding derivatives is to have these first  two necessary conditions in mind:

*

*If $f$ is a derivative on an interval $[a,b]$ then $f$ is a function in the first class of Baire (i.e., a Baire one function).


*If $f$ is a derivative on an interval $[a,b]$ then $f$ is a Darboux function (i.e., $f$ has the intermediate value property, just like a continuous function has).


*Those are necessary, but not sufficient conditions.  To learn more will take some considerable effort.  [See the monograph by Andy Bruckner, Differentiation of Real Functions, https://www.amazon.com/Differentiation-Real-Functions-Crm-Monograph/dp/0821869906  if you are serious about this subject.  Lots to learn.]
If you can remember just the first two statements here then questions of this type are trivial.

If an elementary level question asks you to prove that a function is not a derivative, suspect first of all that it is not a Darboux function.

For the situation here:  If $f'(c) = L$ exists but $f'(x) > L+1$ for all $c<x<c+\delta$ for some $\delta>0$ then $f'$ cannot be a Darboux function.  If it is not a Darboux function it is not a derivative. QED.
