Q:

in what sense can a function be prime?

We all know that numbers can be prime. Can the concept of primality be generalised to functions?
If so, how? What are examples of prime functions and under what circumstances is a function prime?

A:

What kind of functions? Functions on the naturals? Complex numbers? Strings? Or functions on functions? Secondly, the notion of "primality" usually only makes sense once you have defined multiplication on your chosen functions. Things are going to be very different depending on your choices.
For example if $F$ is the ring of functions on $\mathbb{R}$ with pointwise addition and multiplication, then the prime elements of $F$ are precisely those elements in $F$ which have exactly one zero. Why? Those with no zero are units, while those with more than one zero are clearly not prime. (Prove it!)

