An identity function f is a function which does not have any effect: it always returns the same value that was used as its argument.

Formally, if M is a set, we define the identity function id_M on M to be that function with domain and codomain M which satisfies

id_M(x) = x    for all elements x in M.

If f : M → N is any function, then we have f o id_M = f = id_N o f. In particular, id_M is the identity element of the monoid of all functions from M to M.

When choosing M equal to the positive integers, one obtains the identity function Id(n), which is a multiplicative function considered in number theory.