Limits of Rolle theorem

Question

I would like to see a function $f:[a,b]\to\mathbb{R}$ that is differentiable in $(a,b)$ but it is not continuous at least at one of the interval boundary points $a$ or $b$. Can you show me one?

This is a curiosity that would make me to see limits of Rolle theorem, because one of its hypothesis is that the function $f$ has to be continuous in the entire closed interval $[a,b]$, even if it could be differentiable only in the open $(a,b)$.

Thank you.

Answer 1

Consider $f(x) = x$ on $(0, 1]$, and $f(0) = 1$.

Answer 2

Consider $f(x) = \frac{1}{x}$ in $[0, 1]$ and define $f(0) = 0.5$.

Then by the extreme value theorem - which is needed to make Rolle's Theorem work - since $f$ doesn't obtain a maximum, $f$ is not continuous on $[a,b]$.