If the sum of two functions is convex is at least one of them also convex?

Question

I know that the sum of two convex functions $f$ and $g$ is convex, but can we also state the opposite? If the sum $f+g$ is convex, does it mean that at least one of them is convex?

Answer 1

No, not necessarily.

Let $f(x) = \sqrt{x}$ on $[0,1]$ and $x-\sqrt{x}+1$ on $[1,2]$

Let $g(x) = x-\sqrt{x}$ on $[0,1]$ and $\sqrt{x}-1$ on $[1,2]$.

Then $f+g = x$ which is convex but neither $f$ nor $g$ are convex.