Smallest Period?
Definition: A periodic function is defined as being any function with the property that f ( x ) = f ( x + c ) for all x in the domain and some positive constant c .
Do all periodic functions have a smallest positive period?
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2 solutions
Same answer!
An example of such a periodic function without a smallest period is
f ( x ) = { 1 0 if x ∈ Q if x ∈ Q
Clearly, this function is periodic, but you cannot give a definition for its smallest period. Thus, not all periodic functions have a smallest period.
And what is the period of this function?
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p can be made arbitrarily small, which is the point of Kesa's problem.
Probably a good idea to provide the most general definition of a periodic function, which is any function such that
F ( x + p ) = F ( x )
for all x where p is any non-zero constant. So, with the example of the Dirichlet function given above, any rational constant added to a rational or irrational x will result in 1 or 0 , respectively, as before. So, curiously, however any 1 in the Dirichlet function is shifted over to another 1 , all the rest of the function matches up!
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According to this definition (allowing negative periods), no periodic function has a smallest period ;) Maybe you should ask: Do all periodic functions have a smallest positive period?
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Okay, my head hurts now. If we go with the most general definition as described above, then if x ′ = x + p , we have
F ( x ′ ) = F ( x ′ − p )
for all x ′ , so that I can't imagine right now where we could have any periodic function that is self-similar only with a positive translation in x . In other words, I don't quite understand your question.
I'm looking at this problem as a symmetry operation, where "something is left unchanged" after some operation. In this case, the operation being a simple translation. .
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@Michael Mendrin – I don't disagree with your definition of a periodic function in terms of a non-zero period p . But in this case we must define the fundamental period (if it does exist) as the smallest positive period.
An ongoing question to do with functional equations (but not really related to periodic functions) is this .
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I will be glad to write a solution if nobody else does. The solution is short, but it requires a bit of linear algebra. You have to accept the Axiom of Choice (you may want to state that in the problem).
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For widest readership, you should go ahead and post your solution now that apparently will involve the Axiom of Choice.
Consider a constant function: Any positive real number will be a period.