Definition:Sawtooth Wave/Points of Discontinuity

Definition

Let $S$ be the sawtooth wave or inverse sawtooth wave defined as:

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

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

The points $\paren {2 r + 1} \lambda$, for $r \in \Z$, are jump discontinuities.

The values $\map S {\paren {2 r + 1} \lambda}$ may or may not be explicitly defined.

It is a common approach to include one of the endpoints of the interval from $-\lambda$ to $\lambda$, in order to ensure that the domain of $S$ is simply defined.

For the sawtooth wave for example:

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

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

and, for the inverse sawtooth wave:

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

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

Another approach is to make $\map S {\paren {2 r + 1} \lambda} = 0$ for all $r \in \Z$.

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