under what circumstances $\sup_{x \in \mathbb{R}} \{ f(x) + g(x) \} = \sup_{x \in \mathbb{R}} \{f(x) \} + \sup_{x \in \mathbb{R}} \{g(x)\}$?

Question

Let $f$ and $g$ be real-valued functions on $\mathbb{R}$.

As we know that the elementary property of supremum of functions is $$\sup_{x \in \mathbb{R}} \{ f(x) + g(x) \} \leq \sup_{x \in \mathbb{R}} \{f(x) \} + \sup_{x \in \mathbb{R}} \{g(x)\}. $$

I'm curious about under what circumstances the equality will hold? I thought constant functions $f,g$ could make it to be equal. What else?

Any suggestions would be greatly appreciated.

Answer 1

One can be $f,g$ be monotone functions of same type. and It doesn't have to be strictly monotone! This even covers all constant functions

Another one can be only $f$ be a constant function , No condition on $g$

Answer 2

It is equivalent to:

for each $n\in \mathbb Z ^+$: $f^{-1}(\sup(f)-\frac{1}{n},\sup f]\cap g^{-1}(\sup(g)-\frac{1}{n},\sup g ]\neq \varnothing$