# Math Help - Check 1 more Real

1. ## Check 1 more Real

Suppose we have a continuous function f that is on the closed interval [0,1] and has a range that's contained in [0,1] as well. Prove f has to have a fixed point (ie: show $f(x) = x$ for at least 1 value of $x \in [0,1]$.

SOLUTION:

We are done if $f(0) = 0$ or $f(1) = 1$

What about if $f(0) > 0$ and $f(1) < 1$.

Let $g(x) = x - f(x)$, then we know that $g$ is continuous and further $g(0) < 0, g(1) > 0$. By the Intermediate Value Theorem (IVP), there exists a $c \in (0,1)$ with $g(c) = 0$. That is, $f(c) = c$. And hence the proof is complete.

2. Yes, that proof works nicely.