# restriction of a function

## Definition

Let $f\colon X\to Y$ be a function from a set $X$ to a set $Y$. If $A$ is a subset of $X$, then the restriction of $f$ to $A$ is the function

 $\displaystyle f|_{A}\colon A$ $\displaystyle\to$ $\displaystyle Y$ $\displaystyle x$ $\displaystyle\mapsto$ $\displaystyle f(x).$

Some authors write $f\upharpoonright A$ instead of $f|_{A}$.

## Properties

If $A\subseteq X$, and $B\subseteq Y$, then

 $\displaystyle(f|_{A})^{-1}(B)$ $\displaystyle=$ $\displaystyle A\cap f^{-1}(B).$
Title restriction of a function RestrictionOfAFunction 2013-03-22 13:43:05 2013-03-22 13:43:05 yark (2760) yark (2760) 14 yark (2760) Definition msc 03E20 restriction Pullback2 ExtensionOfAFunction