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Feb 24, 2018 at 1:51 comment added jcdornano 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.
Feb 24, 2018 at 1:21 comment added Gerald Edgar 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?
Feb 24, 2018 at 1:15 history closed Igor Rivin
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Duplicate of Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Feb 24, 2018 at 1:07 history asked jcdornano CC BY-SA 3.0