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 $L$ iff $f + k$ is periodic with period $L$.

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

Then:

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

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

Then:

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