On a previous question, Pete L. Clark wonders if there are any applications of $f'(a)=\lim_{x\to a}f'(x)$. Ben mentions one in the coments. I'm wondering if there are any other interesting usages of this, be it for particular functions or for proving other theorems.
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So what are the applications of a continuous first derivative at $x=a$? – Oct 09 '23 at 20:14
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The question isn't being correctly replicated here. You can see it in the link. It is: one can show (I learned this from Spivak's Honors Calculus) that if $I$ is an interval and $f: I \rightarrow \mathbb{R}$ is a function that is defined and differentiable at all points of $I$ except possibly at one point $a$ and $\lim_{x \rightarrow a} f'(x)$ exists, then $f'$ is differentiable at $a$ (hence $f'$ is continuous at $a$). In other words: when does this result help in practice to show that a function is differentiable at $a$? – Pete L. Clark Jul 22 '24 at 15:38