Periodic Function plus Constant

Theorem
Let $f: \R \to \R$ be a real function.

Let $k \in \R$ be constant.

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