# Proof for the property of a fixed point of $f$.

Suppose that $f \colon [a, b] \to [a, b]$ is continuous. (Note that the range of $f$ is a subset of $[a, b]$)

Prove that there exists at least one point $x \in [a, b]$ such that $f(x) = x$. A point with this property is known as a fixed point of $f$.

If $f(a)=a$ or $f(b)=b$ then we are done, suppose $f(a)\ne a, f(b) \ne b$
Now what can you say about $g$ below? does it vanish somewhere in $(a,b)$?
$g(x)=f(x)-x$