Classification of Periodic Functions
1. Notation
For convenience, throughout the article, 
will be real valued function.
2. What we are trying to do?
Consider the set
So 
is collection of all possible values of periods as well as its counterparts (with the opposite sign) with trivial element 
. What we are trying to do is to examine the structure (or elements) of .
3. Observation
If 
, then 
. So is closed under addition.
If 
, then 
.
Combining the facts 1,2, and 3, we can deduce that is additive subgroup of 
. Now it is time to divide cases.
4. Case By Case Examination
4-1.
If 
, then from observation (3), we have 
; so that 
. In this case, 
is non-periodic.
4-2.
Suppose 
and 
(i.e, has fundamental period). Take any strictly decreasing sequence of periods of 
; 
such that 
. Then define another sequence 
. By observation (2) and (3), 
is also period, with properties 
and 
. This contradicts to the fact that 
is infimum, so there should exist some 
such that 
for all 
. This implies 
for all . So 
.
Pick any 
. We can find appropriate integer 
such that 
. Since 
and 
, the only possibility is 
. This shows that
4-3.
Now suppose but 
. Let 
be given. Pick such that 
. Now for any 
, we can find appropriate integer such that 
. Then 
and 
; therefore 
is dense in 
. Similar reasoning shows that 
is dense in 
.
5. Conclusion
Now we've finished the classification. First any real valued function with domain is one of the following.
Class 1. It is non-periodic, which is the case when 
.
Class 2. It is periodic and has fundamental period, therefore any period is just integer multiple of fundamental period.
Class 3. The set of periods is dense subset of .
In Math Talk #1, we proved that every continuous periodic functions has fundamental period, so those continuous periodic functions are all of class 2. Also for a constant function 
, 
, which is the trivial case of class 3. Finally the Dirichlet Function,
which we discussed in
Math Talk #1 - Example 1, has
. Since set of rationals is dense subset of real numbers, it is the first non-trivial case of class 3.
6. End
Well, these are the typical usage of elementary analysis. The study of analysis in mathematics not only helps us to examine mathematical objects rigorously, but also gives us deeper understanding of concepts that we already know.
