Limits of Rolle theorem

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

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 $[a,b]$.