Equality case of triangle inequality with functions If $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that $\left|f(x) + g(x)\right| = \left|f(x)\right| + \left|g(x)\right|$ for all $x \in \mathbb{R}$, then must $f = cg$ for some $c \gt 0$?

You are almost correct. Expanding on Andre's answer... the condition you want is
$$f(x) g(x) \ge 0$$
