$f$ is continuous on $[0,1]$ and the range of $f$ is a subset of $[0,1]$. Prove that there is a number $y \in \mathbb{R}~$ in $[0,1] $ such that $f(y) = y$ using $~h(x) = f(x) - x$
I wasn't sure how to approach this as I couldn't seem to determine whether the $~h(x) = f(x) - x$ was an identity of continuous functions or just part of the question. It seemed to be close to the definition of continuity at a point $\mu$:
If $\lim\limits_{x \to \mu} f(x) = f(\mu)$
but I wasn't sure how to use it (if I should at all).
What is a good direction to take with this?