Question

Use the intermediate value theorem to prove

Suppose f is continuous and 0<=f(x)<=1 for all x in (0,1) then exists at least one c in [0,1] such that f(c) = c

Answer 1

if \(f(0)=0\) you are done, right?

Answer 2

similarly if \(f(1)=1\) yore are done, so you can assume \(f(0)=a, f(1)=b, a>0, b<1\)
then the gimmick is to use the intermediate value theorem on the also continuous function \[g(x)=f(x)-x\]