Is there an example of continuous map $f:\mathbb R\to \mathbb R^2$ such that there exists $a\in \mathbb R$ such that for all $]y,z[\subset \mathbb R$ , $f([y,z])$ has non empty interior and such that for any $x>a$, $f(]a,x])$ is a neighborhood of $f(a)$.

I think that if $f$ sends a n intervalle on a convexe there exists such an $a$. The existence of such a fonction is an open problem discussed heure : Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$)

(We can ask the same question without asking that for all $]y,z[\subset \mathbb R$ , $f([y,z])$ has non empty interior (is it strictly weaker?) , but my motivation is the problem I just spoke about, where this property holds)