I have a question about the corollary to theorem 5.12 in Rudin's Principles of Mathematical Analysis (page 108):
Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f'(a)< \lambda < f'(b)$ then there is a point $x \in (a,b)$ such that $f'(x) = \lambda$
Corollary: If $f$ is differentiable on $[a,b]$ then $f'$ cannot have any simple discontinuities on $[a,b]$.
Can someone help me to show how he uses the result in the "main theorem" in the corollary?
(There are two cases of simple discontinuities $f(x+) = f(x-) \neq f(x)$ and $f(x +) \neq f(x-)$