Is $ \max_{x\in\mathbb{R}^n} \{ f(x)+g(x) \} = \max_{x\in\mathbb{R}^n} f(x)+\max_{x\in\mathbb{R}^n} g(x) $ if $f$ and $g$ are affine in $\mathbb{R}$?

Question

Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$.

Is the following property true? $$ \max_{x\in\mathbb{R}^n} \{ f(x) + g(x) \} = \max_{x\in\mathbb{R}^n} f(x) + \max_{x\in\mathbb{R}^n} g(x) $$ Of course, for arbitrary functions this is $\leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.

Could anyone verify, and possibly sketch a small proof?

Greatly appreciated.

Answer 1

This cannot be true since the maximum of an affine function $f$ on $\Bbb R^n$ is always $+\infty$ unless it's a constant function.