# Definition Given a [[map]] $f: X \rightarrow Y$, the restriction of $f$ restriction to $A \subset X$ is denoted by $f|_{A}:A\rightarrow X$. # Properties 1. Restricting a function to its domain gives back the function $f|_{X} = f$ 2. If $A \subseteq B \subseteq X$ then $(f|_{B})|_{A} = f|_{A}$ 3. The restriction of a [[Continuous Map|continuous map]] is continuous. 4. The restriction of the identity function on a set $X$ to a subset $A$ of $X$ is the [[Inclusion Map|inclusion map]] from $A$ [[Types of Maps|into]] $X$. # Examples The factorial function is the restriction of the [[Gamma function|Gamma function]] to positive integers, with the argument shifted by one. i.e. $ \Gamma|_{\mathbb{Z}^+}(n) = (n-1)! $