Very Interesting Intermediate Value Theorem Question Calculus

Very Interesting Intermediate Value Theorem Question Calculus

Show that if $f$ is continuous on $[0,1]$ and $0\le f(x) \le 1$ for all
$x\in[0,1]$, then there exists one point $c \in [0,1]$ at which $f(c) =
c$.
Hint (Apply Intermediate Value Theorem to the function $g(x) = x - f(x)$.)