Every $f : [a, b] → [a, b]$ has a fixed point and $f$ is continuous (on $[a,b]$). Deduce the intermediate value theorem.
I managed to show the other way, now I'm here.
I know that $f(c)=c$ for some $c\in [a,b]$, and I need to show that for all $x\in [f(a),f(b)]$ there is $f(y)=x$.
I know that $a<f(a),f(b)<b$ and $f(c)=c$.
Can somebody give me a hint?