Restriction of a function

Let $f:X\to Y$ be a function and let $A\subseteq X$. The restriction of $f$ to $A$ is the function

Restrictions are used constantly in analysis to localize statements (e.g., continuity on a subset, behavior near a point, or defining inverses on domains where a function becomes injective).

Examples:

• If $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$, then $f|_{[0,\infty)}:[0,\infty)\to\mathbb{R}$ is injective.
• If $f(x)=\sin x$ on $\mathbb{R}$, then $f|_{[-\pi/2,\pi/2]}$ is bijective onto $[-1,1]$.
• If $A=\varnothing$, then $f|_{\varnothing}:\varnothing\to Y$ is the unique function with empty domain.