Periodicity equation

Summary

The periodicity equation is the functional equation $f(x + p) = f(x)$ for a fixed nonzero number $p$ and an unknown function $f$ on $\R$. Its solutions are exactly the periodic functions with period $p$. It is the simplest linear difference equation with constant coefficients, and many other equations reduce to it. For example, the general solution of $g(x + 1) = g(x) + 1$ is $g(x) = x + f(x)$, with $f$ a solution of the periodicity equation for $p = 1$.