Does differentiability on $(a, b)$ implies continuity on $[a, b]$? Provided that $f$ is defined on $[a, b]$, is it true that the differentiability of $f$ on $(a, b)$ implies that $f$ is continuous on $[a, b]$ ?
I'm asking this question because I'm reading Salas's calculus and, as he goes trough some theorems, it's not clear on which interval (closed or open) the function is defined.
For example, in one of the proofs the following is asserted: $f$ is differentiable on $(a,b)$; let $x_1$ and $x_2$ be any two numbers on $(a,b)$; then f is continuous on $[ x_1$ , $x_2]$. That would seem to be true if $x_1$ is different from $a$ and $x_2$ is different from $b$. And nothing is said explicitly about which values $f$ may take on $a$ and $b$.

Most probably it is meant that $f(a)$ is the limiting value of the continuous $f$ on $(a,b)$ as $x$ tends to $a$. Similarly for $b$.
