Identity functions

Let \(A\) be a fixed subset of \(\mathbb{R}\).

Define \(\mathrm{id}_A\) to be
\[ \mathrm{id}_A(x) = x \]
for all \(x \in A\).

We say that \(\mathrm{id}_A\) is the identity function on the set \(A\).

Intuitively, the identity function on a set \(A\) leaves every element of \(A\) untouched. Some books use the notation \(1_A\) to denote the identity function on the set \(A\).