Periodic Function plus Constant

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Theorem

Let $f: \mathbb F \to \mathbb F$ be a function, where $\mathbb F \in \left\{{\R, \C}\right\}$.

Let $k \in \mathbb F$ be constant.


Then $f$ is periodic with period $L$ if and only if $f + k$ is periodic with period $L$.


Proof

Sufficient Condition

Let $f$ be periodic with period $L$.

Then:

\(\displaystyle f \left({x}\right)\) \(=\) \(\displaystyle f \left({x + L}\right)\) Definition of Periodic Function
\(\displaystyle \implies \ \ \) \(\displaystyle f \left({x}\right) + k\) \(=\) \(\displaystyle f \left({x + L}\right) + k\)

Thus $f + k$ has been shown to be periodic with period $L$.

$\Box$


Necessary Condition

Let $f + k$ be periodic with period $L$.

Then:

\(\displaystyle f \left({x}\right) + k\) \(=\) \(\displaystyle f \left({x + L}\right) + k\) Definition of Periodic Function
\(\displaystyle \implies \ \ \) \(\displaystyle f \left({x}\right)\) \(=\) \(\displaystyle f \left({x + L}\right)\)

Thus $f$ has been shown to be periodic with period $L$.

$\blacksquare$