Does a convex function preserve convexity of sets?

Question

A convex function is defined in a way that’s a bit weird to me, as it’s defined as a function whose area above the graph of a convex set is always convex.

Is it also true that if $f:S\to T$ is convex, then any convex set in $U\subseteq S$ maps to a convex set $f(U)\subseteq T$? This would mean a convex function is a “structure preserving map for structures on which convexity is defined”.

Answer 1

Partial answer. Convex sets in an Euclidean space are connected and convex functions on open sets are continuous. So if $f$ is convex on some open set containing a convex set $S$ then the image is connected, hence also convex in $\mathbb R$. Also the image of any convex set in an Euclidean space (or even a normed linear space) under a real valued continuous convex function is convex. This is because the image is connected and any connected subset of $\mathbb R$ is an interval (which is convex).

Answer 2

An example of a convex function that is not continuous is $f$ defined on $[0,\infty)$ by $$ f(x) = \cases{1 & if $x =0$\cr 0 & if $x > 0$}$$ The range of this function is $\{0,1\}$ which is not convex.