1) Assuming f is a real-valued function defined on [a,b] of the real line, f is differentiable on the whole [a,b], is it possible that $\displaystyle \lim\limits_{x \to a} f'(x)$ does not exist (both finite and infinite) while $\displaystyle f'(a)$ exists?

2) Assuming function $\displaystyle f: E\to\mathbb R$, E a subset of real line, let E' be the subset of E on which $\displaystyle f'$ is defined, is it possible that E' contains isolated point?

Thanks.