Can you always decompose an $f: [0,\infty) \to \mathbb{R}$ into the sum of a monotonic function and a periodic function?

Question

Is there a nice class of functions, like smooth or smooth and bounded, such that you can always decompose an $f: [0,\infty) \to \mathbb{R}$ into the sum of a monotonic function and a periodic function? It seems true to me but I can't find a reference.

An equivalent question is if you can decompose a sufficiently nice odd $g: \mathbb{R} \to \mathbb{R}$ into the sum of a monotonic function and a periodic function.

Answer 1

Consider $f(x)=\sin(x^2)$. That function is smooth (actually analytic), bounded and can't be written as a sum of a monotonic function and a periodic function.