Periodicity equation

Remarks

Example: $f(x) = \cos(\pi x)$ is a solution for $p = 2$, since $\cos(\pi x + 2\pi) = \cos(\pi x)$. Non-example: $f(x) = x^2$ is not a solution for any $p \ne 0$, since $f(0 + p) = p^2 \ne 0 = f(0)$. With $q = \abs{p}$, the general solution is $f(x) = h\big(x - q \floor{x/q}\big)$, where $h \colon [0, q) \to Y$ is arbitrary. Every solution for $p$ is also a solution for each nonzero integer multiple $np$. The analogous equation $f(x + p) = -f(x)$ is called the antiperiodicity equation; its solutions have period $2p$.