Here’s a neat little problem that I learned about during a party in my first year of graduate school. I don’t know where it’s from originally, but I got it from Joe Miller:
Show that there exist two periodic functions whose sum is the identity function:
for all .
Here periodic has the standard definition: a function is periodic if there exists a constant such that for .
Obviously the functions and can’t be continuous, since any continuous periodic function is bounded. Indeed, it is possible to show that and can’t both be measurable.
In case you’re interested, this paper gives a complete criterion for determining whether a function can be written as a sum of periodic functions with specified periods, and this paper investigates the question of whether such functions can be measurable.