Fourier series can be used to approximate a general periodic function, even with discontinuities. It does this by using the sum of continuous sine and cosine waves.
The Fourier series of the periodic function $f(x)$ defined on the domain $[−\pi,\pi]$ such that $f(x)+f(x+2\pi)$ is given by:
$$f(x) = \frac{1}{2}a_0 + \sum_{n=1}^{\infty} \left( a_n\cos{nx} + b_n\sin{nx} \right)$$
where
$$a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$$
and
$$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos{nx}\,dx \textrm{, and} \quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin{nx}\,dx \quad \textrm{for}\, n = 1,2,\ldots.$$
This formula can be extended to functions with an arbitrary period.
In addition, any function defined on an interval $[a,b]$ can be extended to a periodic function, with period $b−a$, for which a Fourier series can be calculated.
For more details on the mathematics underlying Fourier series see https://mathworld.wolfram.com/FourierSeries.html and the references contained within it.