We take $f: [ \alpha, \beta ] \to \mathbb{R}$ a continuous function. We consider $E = \{ P(f), P \in \mathbb{R}[X] \}$ the algebra generated by $f$.
And then, we have to find a necessary and sufficient condition on $f$ so that every continuous function defined on $[ \alpha, \beta ]$ is the uniform limit of a sequence of elements of $E$.
So if I am not mistaken, a necessary and sufficient condition is that the function $[\alpha, \beta] \to \mathbb{R}$, $x \mapsto x$ is the uniform limit of a sequence of elements of E.
But is there better ? Can we find something more simple ?
Thank you.