Discontinuous function

A function $f : X \rightarrow Y$, where $X$ and $Y$ are topological spaces, that is not a continuous function on $X$. The Baire classes, the piecewise-continuous functions and the step functions are important classes of discontinuous real-valued functions $f : X \rightarrow \mathbf{R}$.

Discontinuous functions occur, for example, when integrating elementary functions with respect to a parameter (see Dirichlet discontinuous multiplier), when calculating the sum of a series in which the terms are elementary functions, in particular when calculating the sum of a trigonometric series, and in optimal control problems.

Examples.

$$ \sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ 1+x^2 & \text{otherwise} \ . \end{cases} $$ $$ \sum_{n=1}^\infty \frac{\sin nx}{n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ \frac{\pi-x}{2} & \text{if} \ 0 < x < \pi \ . \end{cases} $$