$\def\R{\mathbf{R}}$ Let $f:[a,b]\to\R$ be a continuous function, that also happens to be differentiable on $(a,b)$. What additional hypotheses must we make to guarantee differentiability on all of $[a,b]$? I first thought the above statement would suffice but the sqrt function, $g(x)=\sqrt{x}$ on $[0,2]$ is a counterexample. My current theory is that by strengthening the hypothesis to say that: if the limit
$$ \lim_{x\to b}{f'(x)}$$ exists, then we can conclude $f$ is differentiable at $b$, and the value of the above limit is indeed equal to $f'(b)$. I don't know how to prove this, but this feels plausible to me. An analogous result should hold for the other endpoint, $a$. Prove or disprove? in the case of disprove, what additional constraints must we make?
