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I have a problem finding an example of a periodic function that does not have a primitive period. Can someone give me such example or explain how to find such function?

Adrian12
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  • A trivial one is a constant function. – Rodrigo Dias Oct 24 '18 at 09:29
  • See also https://math.stackexchange.com/questions/2942016/function-with-arbitrary-small-period – lhf Oct 24 '18 at 11:30
  • You recieved 2 answers to your question. Is any of them what you needed? If so, you should upvote all the useful answers and accept the answer that is most useful to you. – 5xum Oct 25 '18 at 09:12

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One obvious example is the constant function, for which every real number is a period.

Another example would be the function $1_{\mathbb Q}$ (the indicator function for the set $\mathbb Q$), defined as

$$\begin{cases}1 &\text{if } x\in\mathbb Q\\0&\text{otherwise}\end{cases}$$ for which every rational number is a period.

5xum
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The set of all periods of a function is an abelian group. Not every non trivial abelian group has a generator. The rationals under addition is an example with no generator. Consider two real numbers to be equivalent if their difference is a rational number. Any non constant function that is constant on the equivalence classes of this relation will be a periodic function with no primitive period because all rationals are periods. Of course, such a function is not continuous anywhere.

Somos
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