Periodic function

A function is periodic when $f(x+T)=f(x)$ for some nonzero period $T$ and every $x$ in its domain.

For fixed positive integer period $n$, integer-valued periodic functions on $\mathbb{Z}$ are in bijection with ${\mathbb{Z}}^n$ and hence form a countable set. Their union over periods is a countable union of countable sets. In contrast, period-one integer-valued functions on $\mathbb{Q}$ can prescribe arbitrary binary values on its infinitely many distinct classes modulo $\mathbb{Z}$. The power set of a countably infinite set injects into them, proving uncountability.

For an integrable periodic function $f$ with period $T$, its period average is $⟨f⟩=T^{-1}{\int }_{t_0}^{t_0+T}f(t)dt$, independent of $t_0$. In particular a sinusoidal square has period average $1/2$.

A triangular wave is a continuous piecewise linear periodic function whose slope alternates between two values. The even $2\pi$-periodic extension of $bx$ from $[0,\pi ]$ is $P(s)=b|s-2k\pi |$ for $(2k-1)\pi \le s\le (2k+1)\pi$. Its Fourier cosine series contains a constant term and odd modes with coefficients proportional to the inverse square of the mode number. Such extensions describe reflection under Neumann boundary conditions.