Lim inf of sum of functions

Question

When is the lim inf of sum of two real valued functions equal to the sum of their individual lim inf? That is, I am looking for condition on $f$ and $g$ under which $\liminf\limits_{x \rightarrow \bar{x}} (f(x) + g(x)) = \liminf\limits_{x \rightarrow \bar{x}} f(x) + \liminf\limits_{x \rightarrow \bar{x}} g(x).$

Answer 1

One sufficient condition is: $g=\psi\circ f$ where $\psi:\mathbb R\to\mathbb R$ is an increasing function. Indeed, in this case a sequence $x_n$ that satisfies $\lim f(x_n)=\liminf f$ also satisfies $\lim g(x_n)=\liminf g$, and therefore $$\liminf (f+g)\le \lim (f(x_n)+g(x_n))= \liminf f+\liminf g$$ (the converse inequality being true for all $f,g$).

I would not expect any non-tautological necessary and sufficient condition here.