So, I have a continuous function $f$ from $\left[0,1\right]$ to $\left[0,1\right]$ such that : $$\forall \, x\_1,x\_2\in\left[0,1\right]:\; \mid x\_1-x\_2\mid\;\leq \;\mid f(x\_1)-f(x\_2)\mid$$
It's easy enough to prove that $f$ is both surjective and injective so $f$ is a bijection, and also $\{f(1),f(0)\}=\{0,1\}$. Now I'm trying to prove that $f$ must have a fixed point. When drawing an example it seems obvious, but I'm not sure how to go about the proof, setting aside the trivial case when $f(1)=1$ and $f(0)=0$. Since $f$ is a bijection and if we assume we're in the case $f(1)=0, f(0)=1$ then $f$ is strictly decreasing but not sure how that would help. 
So could anyone give some kind of hint or an observation I might have missed ? Thanks.