NL-NWO 1964 Problem 4

The function $ƒ$ is defined at $[0,1]$ , and $f\{f(x)\} = ƒ(x)$ . $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$ . Determine $f$ .

Prove that the function $g$ , with $g(x) = x$ , $0 \le x \le k$ , $g(x) = k$ , $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$ .