Points of discontinuity and convergence

In equation (1) there is a function on the left and the series on the right meaning that the series converges to the function at each point $x \in {\mathbb{R}}$. It often happens that the Fourier series of a function $f$ fails to converge to that function, in particular at the points of discontinuity of $f$. The fact is that if the function $f$ is piecewise smooth (that is, its derivative is piecewise continuous) then its Fourier series converges for every $x$ to the average value

$$f_{av}(x) = \displaystyle{ \frac{f(x+) + f(x-)}{2},}$$ (9)

where $f(x+) = \lim_{t \rightarrow x+} f(t)$ is the right-hand-side limit of $f$ and $f(x-) = \lim_{t \rightarrow x-} f(t)$ is the left-hand-side limit. Note that at the points where $f(x)$ is continuous, $f_{av}(x) = f(x)$. All the functions we shall consider in the sequel are piecewise continuously differentiable, and therefore the Fourier series will represent the function.

In order to ensure that the Fourier series of function $f$ converges to that function at every $x \in {\mathbb{R}}$, sometimes it is necessary to redefine $f(x)$ at the points of discontinuity $x$, so that $f(x) = f_{av}(x)$. In Example 4 we notice that at the points of discontinuity $\pm n\pi, n = 0,1,2, \ldots$ the average value $f_{av} is 0.5$ whereas the value of the function is 1 for $n$ even and 0 for $n$ odd. Thus, we need to redefine the values of $f$ to be $0.5$ at these points.