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 idM on M to be that function with domain and codomain M which satisfies
- idM(x) = x for all elements x in M.
If f : M → N is any function, then we have f o idM = f = idN o f. In particular, idM 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.
