Fourier coefficients for $2\pi$-periodic functions

In 1822, the French scientist Joseph Fourier conjectured that every function of period $2\pi$ can be represented by the series of the form (1), where

$$a_0 = \frac{1}{\pi} \displaystyle{\int_{-\pi}^{\pi}} f(x) dx,$$ (5)

$$a_n = \frac{1}{\pi} \displaystyle{\int_{-\pi}^{\pi}} f(x)\cos nx dx,$$ (6)

$$b_n = \frac{1}{\pi} \displaystyle{\int_{-\pi}^{\pi}} f(x)\sin nx dx.$$ (7)

There is a mild restriction on a function $f$ defined on an interval $I$ under which the above statement is true: it is required to be piecewise continuously differentiable. (This means that there are finitely many points $\{x_i\}$ in $I$ such that $f$ is differentiable except possibly at these points, and the derivative is piecewise continuous. We recall that a function $f(x)$ is called piecewise continuous if it is continuous except possibly at finitely many points, and at each such point of discontinuity $x$, the one-sided limits $\lim_{t \rightarrow x+} f(t)$ and $\lim_{t \rightarrow x-} f(t)$ both exist and are finite. Note that both functions in Example 2 satisfy this requirement.