A Problem About Periodic Functions

[Phillip B]

Here's a nice problem. I already know the answer, unfortunately, but for the rest of you all it should make for a good icebreaker.

Find two periodic functions, f, g from R to R, such that their sum f+g is the identity function. You are allowed the axiom of choice.

Definition of a periodic function: A function, f, is periodic iff there exists p>0 s.t. for all x in R, f(x+p) = f(x).

I'm looking forward to meeting you guys!

[Susam Pal]

Hi Phillip,

Thank you for posting this nice problem. It introduced me to the concept of R as a vector space over Q and Hamel basis.

Here is the solution I learnt while reading blog posts related to this subject: [PDF] [TEX]

I have included the links to the blog posts I read in the "References" section of the document.

Thanks,

Susam

[Phillip B]

Very nice solution! It's quite interesting that this doesn't work for e^x. I wonder if there's any general way to tell for any function.