real analysis - Must a function that maps bounded convex sets (minus straight line segments) to bounded convex sets be continuous everywhere?

This question in the title came to my mind while I was sitting with my granny in front of my house maybe about half an hour ago.

Although it looks innocent I do not know at the moment some simple argument that would prove or disprove it.

So, let us suppose that we have some function $f$ which is defined on the whole $\mathbb R^{n}$ and that if $S \subset \mathbb R^{n}$ is bounded convex set which is not the straight line segment (with or without boundary points) that then $f(S)$ is also bounded convex set.

Is every such $f$ continuous everywhere?

Be aware that we do not consider functions defined on $\mathbb R$ because bounded convex sets on $\mathbb R$ must be straight line segments (with or without boundary points) so we seek to find an answer for $n \geq 2$.