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 $x=a$? – Annika Oct 09 '23 at 20:14