Function as sum of two periodic functions

Question

Can an arbitrary function from the real numbers to the real numbers can be expressed as the sum of two periodic functions?

Answer 1

Take the example of $\exp(x)$. Suppose that $\exp(x)=f(x)+g(x)$, with $f$ periodic of period $u>0$, and $g$ periodic with period $v>0$.

Then we have $\exp(x+u)=f(x)+g(x+u)$ and hence $\exp(u)(f(x)+g(x))=f(x)+g(x+u)$.

This show that $(\exp(u)-1) f(x)=g(x+u)-\exp(u)g(x)$ and hence as $g(x+u)$ and $g(x)$ are periodic with period $v$, $f$ is periodic with period $v$.

In the same way, $g$ is periodic with period $u$, and $\exp(x)$ is periodic with period $u+v>0$, contradiction.