Can we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?
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4$\begingroup$ Yes, and anything. $\endgroup$– user296602Aug 7, 2017 at 7:05
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$\begingroup$ See Sassatelli last paragraph. A constant function is a somewhat trivial example of a periodic function with any period. $\endgroup$– Peter SzilasAug 7, 2017 at 7:34
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$\begingroup$ 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 ? $\endgroup$– reunsAug 7, 2017 at 7:38
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$\begingroup$ Why was this closed? It was stated as being off-topic, but in my opinion is most certainly on-topic. $\endgroup$– MattHuszJul 14, 2021 at 14:28
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3 Answers
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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.
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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.
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$\begingroup$ 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. $\endgroup$ Aug 7, 2017 at 7:17
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$\begingroup$ 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. $\endgroup$ Aug 7, 2017 at 7:24
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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.
