Intermediate Value Theorem

Homework Statement

Let f : [0; 1] -> [0; 1] be continuous on [0; 1]. Prove that there exists C $$\epsilon [0; 1]$$ such
that f(c) = c.

The Attempt at a Solution

I've manage to prove this by having an extra cont. function g(x)=f(x)-x .. But im looking for easier proof