Period of two periodic functions with same period

Question

Let's say I have a two periodic functions f(x) and g(x) each with the same period of p. Is it always the case that the sum of these two functions will also have the period of p? Is there any counter example?

Answer 1

If you define period of a function $h$ as the number $p$, such that $h(x + p) = h(x)$, for all $x$, then yes (try it).

If you define period as the smallest positive number, such that $h(x + p) = h(x)$, then no, for example: $g = \sin$ and $f = -\sin$ will give you $h(x) = \sin(x) - \sin (x) = 0$.

Answer 2

For sums and products: Let $p$ be a period of $f(x)$ and let $q$ be a period of $g(x)$. Suppose that there are positive integers $a$ and $b$ such that $ap=bq=r$. Then $r$ is a period of $f(x)+g(x)$, and also of $f(x)g(x)$.

So, if $f(x)$ has $5\pi$ as a period, and $g(x)$ has $3\pi$ as a period, then $f(x)+g(x)$ and $f(x)g(x)$ each have $15\pi$ as a period. However, even if $5\pi$ is the shortest period of $f(x)$ and $3\pi$ is the shortest period of $g(x)$, the number $15\pi$ need not be the shortest period of $f(x)+g(x)$ or $f(x)g(x)$.

And as @Antoine says, let $f(x)=\sin x$, and $g(x)=-\sin x$. Each function has smallest period $2\pi$. But their sum is the $0$-function, which has every positive number $n$ as a period!