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Identity function

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.