Let $(p_n)_{n \in \mathbb N} \subset (0,\infty)$ be a sequence of distinct positive numbers such that their series of reciprocals diverges. By the theorem of Müntz-Szasz the closure of the span of the system $\{ x \mapsto x^{p_n} : n \in \mathbb N\}$ with respect to the supremum norm is equal to $C[0,1]$, the space of continuous functions. I was wondering if an extension of this result of the following form is possible:
Suppose that $f:[0,1] \to (0,\infty)$ is a smooth function which is strictly decreasing (or increasing). Then the closure of the span of system $\{ x \mapsto f(x)^{p_n} : n \in \mathbb N\}$ with respect to the supremum norm is equal to $C[0,1]$.
If this is not the case, can we strengthen the conditions on $f$ so that a statement of the above type holds?