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 $c \gt 0$?

No. Consider $f(x)=x^2$, $g(x)=x^4$. $f+g$ is always positive, and so are $f$ and $g$. So the equality holds, but your condition is false.
