$f$ is strictly increasing, $f (a) < a$, $ f(b) > b$. Number of solutions to $f(c) = c$.

Question

Let $f$ be a strictly monotonic continuous real valued function defined on $[a, b]$ such that $f (a) < a$ and $f(b) > b$.

My claim is that, $f(c)=c$ for $c \in (a,b)$, their is either exactly one such $c$ or infinitely many such $c's$.

How should I prove this?

Answer 1

Hint

If your claim was true, some polynomials of degree three would have an infinite number of roots.