Definition:Square Wave/Points of Discontinuity

Definition

Let $S$ be the square wave defined as:

$\forall x \in \R: \map S x = \begin {cases}

\delta + \gamma & : x \in \openint \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \openint {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - 2 \lambda} & : x > \alpha + \lambda \end {cases}$

The points $\alpha + n \lambda$, for $n \in \Z$, are jump discontinuities.

The values $\map S {\alpha + n \lambda}$ may or may not be explicitly defined.

It is a common approach to include either endpoint of the intervals from $\alpha$ to $\alpha + \lambda$, and from $\alpha - \lambda$ to $\alpha$, in order to ensure that the domain of $S$ is simply defined, for example:

$\forall x \in \R: \map S x = \begin {cases}

\delta + \gamma & : x \in \hointr \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \hointr {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - \lambda l} & : x \ge \alpha + \lambda \end {cases}$

Another approach is to make $\map S {\alpha + n \lambda} = \delta$ for all $n \in \Z$.

The precise treatment of the discontinuities is often irrelevant or immaterial.