Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing function.
Let $a
My attempt at a proof is as follows. Let $c:=\sup\{x:a\leq x\leq b\text{, }x\leq f(x)\}$.
This is where I'm stuck. Since I can't use more powerful theorem such as the IVT I find this problem far more complex.