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.