If $f(x)$ is continuous on $[0,1], \text{ and } 0\leq f(x)\leq1, \forall x \in [0,1], \text{ prove } \exists t \in [0,1] \text{ s.t. } f(t) = t$

Question

My thinking is that $f(x)$ has to intersect with function $g(x) = x$ at some point, but I don't know how to prove this.

Answer 1

Guide:

$f$ is continuous, $g$ is continuous, hence $h=f-g$ is continuous.

Argue that $h(0)$ is nonnegative and $h(1)$ is nonpositive and you can use intermediate value theorem.