1968 IMO Problems/Problem 5
Let be a real-valued function defined for all real numbers such that, for some positive constant , the equation holds for all .
(a) Prove that the function is periodic (i.e., there exists a positive number such that for all ).
(b) For , give an example of a non-constant function with the required properties.
(a) Since is true for any , and
We have: Therefore is periodic, with as a period.
(b) when for some integer , and when for some integer .