Can you prove differentiabilty, by showing that the limit of the derivative exists?

Question

Suppose that for a continuous function $f(x)$ the derivative is well-defined for all $x\neq a$. My question is that if the limit $\lim_{x\to a}f'(x)=L$ exists, would that mean that $f(x)$ is differentiable in $a$ and $f'(x)=L$? And if that limit does not exist, would that imply that the function is not differentiable in $a$?

I found this question, which states the same problem, but I didn't fully understand whether this means that you can use the limit to show differentiability: Proving differentiability at a point if limit of derivative exists at that point

Answer 1

As you already know, from the post that you have mentioned, if $\lim_{x\to a}f'(x)=L$, then $f$ is differentiable at $a$ and $f'(a)=L$. On the other hand, consider the function$$\begin{array}{rccc}f\colon&\Bbb R&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x^2\sin\left(\frac1x\right)&\text{ if }x\ne0\\0&\text{ if }x=0.\end{cases}\end{array}$$Then $f$ is differentiable at $0$ (and $f'(0)=0$). However, the limit $\lim_{x\to0}f'(x)$ doesn't exist. So, the condition “the limit $\lim_{x\to a}f'(x)$ exists” is sufficient, but not necessary, to ensure that $f$ is differentiable at $a$.

Answer 2

Let $A$ be an open set of $\mathbb R$, and let $f:A\to\mathbb R$ be a function.

If $f$ is differentiable at $a$, then this implies that $f$ is continuous at $a$. However, it may well be the case that $f'$ is discontinuous at $a$. In general, there are three kinds of discontinuity that a function $g$ can have at a given point $a$ in its domain:

• If $\lim_{x\to a}g(x)$ exists in $\mathbb R$ but is unequal to $g(a)$, then $g$ has a removable discontinuity at $a$.
• If $\lim_{x\to a^-}g(x)$ and $\lim_{x\to a^+}g(x)$ both exist in $\mathbb R$ but are unequal, then $g$ has a jump discontinuity at $a$.
• If $\lim_{x\to a}g(x)$ does not exist in $\mathbb R$, then $g$ has an essential discontinuity at $a$.

As explained in the linked post, is impossible for $f'$ to have a removable discontinuity or a jump discontinuity at any point. However, it may have an essential discontinuity. That being said, it is impossible for $f’$ to be discontinuous at every point of $A$; this is an advanced result discussed here in more detail.