Prove f has a fixed point if it is increasing but not necessarily continuous

Question

Suppose $f : [0, 1] \rightarrow [0, 1]$ is increasing (but not necessarily continuous).
Show that there is a number $x \in [0, 1]$ with $f(x) = x$.
(Hint: You can’t apply the IVT directly because the function need not be
continuous. Draw a picture and try to copy the proof of the Intermediate
Value Theorem.)

I don't understand how copying the IVT and connecting it to my graph would help me prove this since the IVT has nothing to do with increasing functions(or am I wrong about this?)

Answer 1

Hint

What about $$c=\inf\{x\in [0,1]\mid f(x)\leq x\}$$ if it exist ?