if $h$ is a continuous function which is differentiable at all points except "possibly" the point $a$. Then how come $\lim_{x\to a}h'(x)=b$ and then $h$ is differentiable also at $a$ and $h'(a)=b$?
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The question isn't totally clear (please make sure the body of the question is complete independent of the title); but I don't think the assertion is correct. Consider $h(x) = x^2\sin\frac1x$ when $x\ne0$ and $h(0)=0$. – Greg Martin Dec 01 '21 at 16:53
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1Did someone really say "$\lim_{x\to a}h'(x)=b$ and then ..." or was it "if $\lim_{x\to a}h'(x)=b$ then ..."? – David K Dec 01 '21 at 16:59