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Let $f:\mathbb{R}\rightarrow \mathbb{R}$, where $f(x)=1$ if $x$ is rational number and $0$ if $x$ is irrational number. Is $f$ a periodic function.

A hour ago, in this post, I said that this function is not periodic, then Kenny Lau told that I am wrong. He also said that "well, any rational number is a period... $f(x+r)=f(x)$ if $r\in\mathbb{Q}$".

Now, I am totally confused about periodic functions. It means there must infinitely many periods.

It would be very helpful if someone explain it elaborately.

MAN-MADE
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  • One approach (as in this post) is to assert that such an $f$ is "periodic, but with undefined fundamental period". – Ben Grossmann Aug 30 '17 at 04:04
  • But if the function is periodic, not constant and has at least one point where it is continuous, there is a minimal period, and every period is a multiple of that minimal period. –  Aug 30 '17 at 04:41

1 Answers1

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A periodic function doesn't necessarily have a minimal period. As long as it repeats itself ($\exists c>0: \forall x \in \Bbb R: f(x+c) = f(x)$), then it is periodic.

Kenny Lau
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