The identity function is about the most boring nonconstant function possible:
I(x)=x
(note you should probably also say I:X->X to make sure your domain is OK).

So why bother? Convenience, as always.

Function composition is an associative binary operation; the identity functions are its units (composing an identity function on either end does nothing).

When X is a vector space, the identity function is a linear transformation. Its representation as a matrix is, of course, the identity matrix. More generally, in any group G acting on a set X, the function x|->e*x (where e∈G is G's identity element) is the identity function.

It's unavoidable. So we may as well give it a name...

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