Has the following concept ever been studied/have a standard name?
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided $$f\left( \frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$$ for all $|x-y|\geq 1$.