# 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 $\mathbb{R}$?

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.

• Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist? – Hermione Dec 8 '18 at 11:44

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.
• This is actually very helpful. My specific case is $\max_\nu \{ \langle b,\nu \rangle + \langle \nu, u \rangle + \Vert u \Vert \}$. So this is only equal to $\max_\nu \langle b,\nu \rangle + \max_\nu \{ \langle \nu, u \rangle + \Vert u \Vert \}$ if I set the condition that $\langle \nu, u \rangle = - \Vert u \Vert$, a constant. Correct? – ex.nihil Dec 8 '18 at 11:59