Difficult to understand why Brouwer's fixed-point theorem holds

Question

I am asked to prove or give a counterexample to the following statement:

If $f$ is continuous on the interval $[0, 1]$ and if $0 \leq f(x) \leq 1$ for every $x \in [0, 1]$, then there exists a point $c \in [0, 1]$ such that $f(c) = c$.

I have difficulty understanding the solution which agrees with the statement, since for example if we have $f(x) = 0.5$, this function seems to match every condition?

Answer 1

To respond directly to your proposed wonder:

I have difficulty understanding the solution which agrees with the statement, since for example if we have $f(x)=0.5$, this function seems to match every condition?

If you have a function $f$ continuous on $[0,1]$ and defined by $x \mapsto 0.5$, then there is a point $c \in [0,1]$ with the desired property. In particular, consider $c = 0.5$. Then, you have $f(c) = c$ since $f(0.5) = 0.5$.

This means that your example is not a counterexample.