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The fact that a differentiable function must be continuous is well known, and the fact that a continuous function need not be differentiable is also well known. However, if I say that $f(x)$ and $f'(x)$ are both continuous at $x = a$ then is $f(x)$ differentiable at $x = a$. I can't find a rigorous proof of this, nor a counter example . Providing either would be helpful

MNIShaurya
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    Proof: Since $f'$ is continuous at $a$, then, in particular, $f'(a)$ exists, which means that $f$ is differentiable at $a$. – José Carlos Santos Aug 29 '20 at 15:16
  • You mean perhaps that if $f'(x)$ has a limit when $x\to a$ and $x\neq a$ then $f$ is differentiable at $a$. This is true, if $f'$ exists in a punctured open interval centered at $a$ and $f$ is continuous at $a$. – Gribouillis Aug 29 '20 at 15:24

2 Answers2

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The statement $f’(x)$ is continuous at $x = a$ already implies that $f’(a)$ exists, by definition of continuity. So $f$ is differentiable at $x = a$.

Ken Hung
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Continuously differentiable means that the derivative exists at any point and that the function $f'$ is continuous.

Bellem
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