I have the following problem (Zorich 5.3.7(b)): let $f \in C(I)$, $f$ have derivative on $I$ (so $f'$ exists on $I$, but might be discontinuous in general) $[a, b] \subset I$ and $f \in C^{(2)}((a, b))$. Prove that $\exists \xi \in (a, b)$, such that $f'(b) - f'(a) = f''(\xi) \cdot (b - a)$
I cant use mean value theorem here, because for mean value theorem we need $f'$ be continuous at $[a, b]$, but since it has second derivative only in $(a, b)$, than $f'$ must be continuous only at $(a, b)$ and might be discontinuous at corner points
I tried to rewrite proof of mean value theorem for $g(x) = f'(x) - \frac{f'(b) - f'(a)}{b - a} \cdot (x - a)$, but havent succed here. I can say that $g$ is also continuous on $(a, b)$, but I cant say that it reach it maximum, and minimum (I need that to find point $\xi$ s.t. $g'(\xi) = 0$)
I guess this statement should be corollary of Darboux's theorem (https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) ), but cant figure out how to obtain initial statement from Darboux's theorem. I can say that $f'$ continuous at point $a$ or $\exists \epsilon$ s.t. $f'(a) \in [f'(a + \epsilon), f'(b)]$ and if we get first scenraio everything good, we can use MeanValue, but if we in second case I cant obtain initial statement