# Is a constant function periodic? [closed]

Can we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?

• Yes, and anything.
– user296602
Aug 7, 2017 at 7:05
• See Sassatelli last paragraph. A constant function is a somewhat trivial example of a periodic function with any period. Aug 7, 2017 at 7:34
• You just deleted a similar "question". The question was : what do you want to show really ? Is it the Fourier series vs the Fourier transform ? Aug 7, 2017 at 7:38
• Why was this closed? It was stated as being off-topic, but in my opinion is most certainly on-topic. Jul 14, 2021 at 14:28

Definition 1: For $$P>0$$, a function $$f:\Bbb R\to\Bbb R$$ is periodic of period $$P$$ (or $$P$$-periodic) if $$\forall x\in\Bbb R,\ f(x+P)=f(x)$$ A function is commonly said to be periodic if there is some $$P>0$$ such that $$f$$ is $$P$$-periodic.

Definition 2: For a periodic function $$f$$, the fundamental period of $$f$$ is, if existing, the least positive real number $$T$$ such that $$f$$ is $$T$$-periodic. Specifically, if we call $$L=\inf\{ P>0\,:\,\forall x\in\Bbb R,\ f(x+P)=f(x)\},$$ then $$f$$ has a fundamental period if and only if $$L>0$$; in which case, $$L$$ is the fundamental period of $$f$$.

Answer to your question: Constants are periodic functions of any period and, therefore, they do not have a fundamental period.

Nowhere in the definition of a period function is it stated that the function must have a least period.

If $f(x) = c$ then for any $p$ we have $f(x+p) = f(x)$. So $f$ is periodic and $p$ is a period. Obviously any other non-negative value will also be a period.

There is nothing in the definition of periodic function that says that is not allowed.

• Ah... G. Sassatelli beat me by a minute with a good definition. The idea of a least period simply called the "fundamental period". It does not exist for constant functions. Aug 7, 2017 at 7:17
• For clicks and gigles let $f(x) = 0$ if $x$ is rational. And $f(x) = 1$ if $x$ is irrational. Then for any rational $q$, $f(x+q) = f(x)$ so $f(x)$ is also a periodic function that has no fundamental period. But $f(x) = c$ is the only periodic continuous function that has no fundamental period. FYI. Aug 7, 2017 at 7:24

Yes, a constant function is a periodic function with any T∈R as its period (as f(x)=f(x+T) always for howsoever small 'T' you can find).

However, the fundamental period of a constant function is not defined for the above reason.