Gluing lemma for convex functions?

Question

Let $A, B$ are two open intervals of real numbers such that $A\cap B\neq\emptyset.$ Suppose there are two convex functions $f: A\to\mathbb{R},$ and $g: B\to\mathbb{R}$ satisfying $f(x)=g(x)$ for all $x\in A\cap B.$ Then the function $$ F(x) = \begin{cases} f(x), & \text{if $x\in A\setminus B$} \\ g(x), & \text{if $x\in B$} \end{cases}$$ ought to be convex. If both $f, g$ are differentiable (or twice differentiable) the proof is straightforward. I suppose a proof without this assumption can be given via the three slopes rule, but did not succeed yet. Can somebody give a proof or a reference for this fact.

Answer 1

The statement is true for open intervals $A, B$ because convexity is a local property for functions defined on open intervals.

Every $x \in A \cup B$ has a neighborhood $U$ such that $\left.F\right|_U =\left.f\right|_U$ or $\left.F\right|_U =\left.g\right|_U$, so $F$ is locally convex and therefore convex.