Exercise 12, § 2

Let $f$ be a differentiable real function on an open interval I, g its derivative on I, and $[a, b]$ a compact interval contained in I; suppose that g is differentiable on the open interval ]$a, b[$ but not necessarily right (resp. left) continuous at the point $a$ (resp. $b$); show that there exists $c$ such that $a < c < b$ and that

(use exerc. 4 c) of I, p. 36).