Let $f$ be a continuous function from [a, b] to [a, b], and is differentiable on (a, b). We will say that point y $\in$ [a, b] is a fixed point of f if $y = f(y)$. If the derivative $f'(x) \neq 1$ for all x $\in$ (a, b), then f has exactly one fixed point in [a,b].
I don't understand why. All I get is that if we take two points inside [a,b], there can be no fixed point since the derivative cannot be equal to one. But if we take one point inside the interval and another outside it, can there not be infinite such fixed points?