  We characterize all compact and Hausdorff spaces $X$ which satisfy that for
every multiplicative bijection $\phi$ on $C(X, I)$, there exist a homeomorphism
$\mu : X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$\phi
(f) (x) = f(\mu (x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This
allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134}
(2006), 1065-1075). Some related results on other semigroups of functions are
also given.
