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

Proof

Sufficient Condition

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

Then:

\(\ds \map f x\)

\(=\)

\(\ds \map f {x + L}\)

Definition of Periodic Real Function

\(\ds \leadsto \ \ \)

\(\ds \map f x + k\)

\(=\)

\(\ds \map f {x + L} + k\)

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

$\Box$

Necessary Condition

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

Then:

\(\ds \map f x + k\)

\(=\)

\(\ds \map f {x + L} + k\)

Definition of Periodic Real Function

\(\ds \leadsto \ \ \)

\(\ds \map f x\)

\(=\)

\(\ds \map f {x + L}\)

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

$\blacksquare$

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