Yet another common false belief is : 

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

> Given any two positive real numbers $p_1,p_2$, there exists functions
> $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period
> $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive
> period $=p_2$, such that $f_1+f_2$ is also periodic.