Periodic Function plus Constant

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 $L$.

$\blacksquare$