According to the statement of the Rolle's theorem in Apostol calculus 1, we need to have a continuous function on $S = [a, b]$, and this function should have a derivative on the interior of $S$. I do not get this condition.
a) Why is the derivative restricted to the interior? Doesn't the right derivative ensure the right continuity?
b) As a result, the proof ensures that there is some $c$, s.t. $a < c < b$, where $f'(c) = 0$. However, why not try to prove $a \leq c \leq b$?
Historical part of the question: In Apostol Rolle's theorem is used to prove the mean-value theorem, which is in turn used to prove convexity properties of derivatives, and there is a big problem with endpoints: suppose the derivative 𝑓′(𝑥) is strictly positive on (𝑎,𝑏), then the function is strictly increasing on [𝑎,𝑏]. This is the conclusion from the Rolle's -> mean-value theorems above in Apostol next section. But the Rolle's theorem does not specify the endpoints 𝑎 and 𝑏 as valid places for the derivative zero! It feels unproven that the function 𝑓(𝑥) is increasing on [𝑎,𝑏], when it can be for example decreasing at a point 𝑎, and further increasing on the interior.