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 ?

Question

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  ?

kregde84 · Accepted Answer

I think the answer is yes. Let   f  ∈  C  (      R    ,      R    ) be periodic and not constant. Then there is a minimal positive period   r  &amp;gt;  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  &amp;gt;  0. Then   r  ≤  s. We show that   s  =  m  r for some   m  ∈      N  : Otherwise  ∃  m  ∈      N    :  m  r  &amp;lt;  s  &amp;lt;  (  m  +  1  )  r  .Then   0  &amp;lt;  s  −  m  r  &amp;lt;  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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