Main Page | See live article | Alphabetical index An identity function f is a function which doesn't 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. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.