Can the sum of two periodic functions over $\mathbb R$ be strictly increasing?

Question

If there exists a direct sum decomposition for the real space $\mathbb R=V\oplus W$, and define $f(v+w)=v$, $g(v+w)=w$, then $h(v+w)=v+w$ is strictly increasing. But I can't find explicitly one such decomposition.

Answer 1

The existence of such a decomposition depends on the axiom of choice. It can't be done explicitly.

With the axiom of choice, $\mathbb{R}$ has a Hamel basis $B$ over $\mathbb{Q}$. Separate $B$ into two nonempty pieces; their spans will be the $V$ and $W$ we seek.

See also this old AoPS thread for more discussion on what functions can and can't be expressed as a sum of finitely many periodic functions on $\mathbb{R}$. (Polynomials yes, exponentials no)