Everywhere discontinuous and periodical function should has no minimum positive period?

Asked Jan 10 '24 at 07:43

Active Jan 10 '24 at 07:43

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Let $f$ be an everywhere discontinuous and periodical function, can we justify that $f$ has no minimum positive period? If yes, provide the proof; if no, give an example.

Take the Dirichlet funcition, we see that there exists an everywhere discontinuous and periodical function, who has no minimum positive period. And this let me ask the question above...

Notice that the question is an arbitrary everywhere discontinuous and periodical function! I have no idea, since I just know Dirichlet funcition as an everywhere discontinuous function....

asked Jan 10 '24 at 07:43

xldd

3,785

https://math.stackexchange.com/a/1000475/568973 Nice answer and nice comments below the answer – G. Gare Jan 10 '24 at 07:56
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No, of course not. Take $f(x)={x} d({x})$ where $d$ is the Dirichlet function and ${x}=x-\lfloor x\rfloor$. – Sassatelli Giulio Jan 10 '24 at 07:57
@SassatelliGiulio the $f$ is continuous at $x=0$? $f(x)={x}, x\in \Bbb Q$, and $f(x)=0, x\not\in \Bbb Q$. – xldd Jan 10 '24 at 08:35
@xldd No, it isn't continuous in $0$, and yes to the other two observations. – Sassatelli Giulio Jan 11 '24 at 16:07

Everywhere discontinuous and periodical function should has no minimum positive period?

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