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Periodicity#

Definition: Periodic Function

A real function \(f: D \subseteq \mathbb{R} \to \mathbb{R}\) is periodic iff there exists a \(P \ne 0\) such that

\[ f(x + P) = f(x) \qquad \forall x \in D \]

Definition: Antiperiodic Function

A real function \(f: D \subseteq \mathbb{R} \to \mathbb{R}\) is antiperiodic iff there exists a \(P \ne 0\) such that

\[ f(x+P) = -f(x) \qquad \forall x \in D \]

Theorem: Modifications of Periodic Functions

If \(f(x)\) has a period of \(P\), then \(f(\alpha x)\), where \(\alpha \in \mathbb{R}\) has a period of \(\frac{P}{\alpha}\), provided that \(\alpha \ne 0\).

Proof

TODO