Mean Value Theorem ( i guess )

Question

Suppose that a function $f$ is continuous on the closed interval $[0,1]$ and that $0\leq f(x)\leq 1$ for every $x \in [0,1]$. Show that there must exist a number $c$ such that $f(c)=c.$

Answer 1

$Hint$: remember that when you have a continuos function defined on an interval, you can apply the intermediate value theorem. if $0 \leq f(x) \leq 1$ for each $x\in [0, 1]$, what can you say about the function $h: [0, 1] \rightarrow [0, 1]$ defined by $h(x) = f(x) - x$?