Exercise 1, § 4

Let T and X be two compact spaces, $\pi$ a continuous mapping of T into X, and $g$ a continuous and finite numerical function defined on T. For every $x \in X$, let $f(x)$ be the infimum of $g(t)$ on the set $\pi^{-1}(x)$. Give an example where $f$ is not continuous. (Take $T = [0,1]$ in $\mathbf{R}$, and for $\pi$ the canonical mapping of T onto the quotient space X obtained by identifying 0 and 1 in T.)