Is there a name for a function with the property f(xy) = f(x)f(y)?

If the domain is the integers, then the function is a totally multiplicative function.

Interesting features:

f(1) = f (1 * 1) = f(1) * f(1), so f(1) = 0 or f(1) = 1

If f(1) = 0, then f(x) = f(1 * x) = f(1) * f(x) = 0 * f(x) = 0, so the function is identically zero.

If f(1) = 1 and the domain is real numbers:

If x != 0, then

f(x) * f(1/x) = f(x * 1/x) = f(1) = 1, so f(x) != 0 for x != 0

1 = f(1) = f(x * 1/x) = f(x) * f(1/x) for all x, so f(1/x) = 1/f(x) for x != 0

this means that f(a/b) = f(a)/f(b) for b != 0

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Indeed, a function with this property is called a completely multiplicative function. Often, it is used in integer functions i.e. functions with the (sub)set of integers as the domain. A simple one is the function f defined as f(even integer) = 0 and f(odd integer) = 1. Then f(xy) = f(x)f(y).

"Multiplicative" is indeed one term for such a function.

"Operation preserving" is another (perhaps more widely used) term, or we can say that the function "preserves multiplication" if the specific operation it preserves is considered important.

In general, "f preserves multiplication" is probably the best answer I can give to your question, I think.

Group and ring homomorphisms do have this property also, but they have to satisfy other things as well (the function's domain and codomain must be groups under multiplication), so calling such a function a "homomorphism" is not necessarily correct.

Not a function type, but a term for being able to partition a function like that. "Separation of Variables" Check wiki article on that.

Addendum: I cede to the other explanations given below, since the subject is about number and group theory.

Scythian's comment is good. If you have a partial differential equation under certain conditions you get what's called 'separable' functions.

Shows up in electrostatics, steady state heat flow...

also, in statistics for a joint density function, it's a necessary condition for independence.

In abstract algebra, we call functions of this type homomorphisms.

yes, it is a homomorphism of groups with the multiplication operation on the reals. .

You're right; a link is below.

It looks like the DISTRIBUTIVE LAW.

Guido