Suppose that #f:RR->RR# has the properties #(a)# #|f(x)| le 1, forall x in RR# #(b)# #f(x+13/42)+f(x)=f(x+1/6)+f(x+1/7), forall x in RR# Prove that #f# is periodic?
2 Answers
We are given
Substituting in
Adding these equations and canceling the terms which appear on each side, we get
Next, we can substitute
Repeating this process three more times results in the equation
Next we substitute in
Repeating this process five more times results in the equation
The above equation implies that the difference between
See below.
Explanation:
Here
We have
so calling
Also considering
making
and now we can write
or
but
so
solving for
so