Consider the space R^R of functions from the set of real numbers to the set of real numbers. A real number r is said to be a period of a given function f iff for all real numbers x, f(x+r)=f(x). I define a periodic set to be the set of periods of a given function. I define a continuous periodic set to be the set of periods of a given continuous function. It is easy to show that R itself and also r*Z are continuous periodic sets, where the latter denotes the product of the real number r with the set of integers. Is the converse true? That is, is every continuous periodic set either R or r*Z?

Marcus Bass

Marcus Bass

Answered question

2022-09-18

Consider the space R R of functions from the set of real numbers to the set of real numbers. A real number r is said to be a period of a given function f iff for all real numbers x, f ( x + r ) = f ( x ). I define a periodic set to be the set of periods of a given function. I define a continuous periodic set to be the set of periods of a given continuous function. It is easy to show that R itself and also r Z are continuous periodic sets, where the latter denotes the product of the real number r with the set of integers. Is the converse true? That is, is every continuous periodic set either R or r Z ?

Answer & Explanation

kregde84

kregde84

Beginner2022-09-19Added 10 answers

I think the answer is yes. Let f C ( R , R ) be periodic and not constant. Then there is a minimal positive period r > 0. Otherwise there would be a sequence of positive periods ( r n ) with r n 0. Then f ( R ) f ( [ 0 , r n ] ) for all n and as f is continuous ist would be constant f ( 0 ), a contradiction. Now let s be any period, w.l.o.g. s > 0. Then r s. We show that s = m r for some m N : Otherwise
m N : m r < s < ( m + 1 ) r .
Then 0 < s m r < r, ans since s m r is a positive period this contradicts the minimality of r. Thus any positive period is in r N and so the set of all periods is r Z .

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