Let $f$ be a continuous fonction from $\mathbb R$ to $\mathbb R^2$, such that for any $a<b\in \mathbb R,\,\, f([a,b])$ is convex.
Is there a line $D\subset \mathbb R^2$ such that $f(\mathbb R)\subset D$ ?
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The linked "duplicate" does, indeed, have two answers. However, neither of them answers the question. But perhaps one of the comments does provide a reference for an answer? – Gerald Edgar Feb 24 '18 at 01:21
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I didn't read the related articles yet, but I'm quite happy with the result announced in the comment of Jairo Bochi. If you ask only $f([0,t])$ to be convex for all $t\in \mathbb R_+$, then there is a counterexample! I also like answers but I have to read the related articles before saying something about the fuzzy direction and ideas that are coming to me right now... I will complete this comment If I find relevant observations from this reading. – jcdornano Feb 24 '18 at 01:51