TITLE: Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
QUESTION [67 upvotes]: Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in $\mathbb{R}^2$ which contains the image $f(\mathbb{R})$ of $f$?
Yes, this question seems rather elementary, but I have already spent (or lost?) too much time on this devilish problem, and I have communicated this question to sufficiently many people to know that it is far from trivial...

REPLY [3 votes]: I asked [a very similar question] (convexity of images of space-filling curves) here once.
Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$.  For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \rbrace$ be convex?  All $t$?  Only countably many $t$?  If so, which countable sets?  Topologically discrete ones?  Dense ones?
Perhaps Pietro Majer's answer to that question will shed some light on this one as well.