Let $f\colon X \to Y$ be a function and $A$ and $B$ be sets such that $X\subseteq A$ and $Y\subseteq B$ An extension of $f$ to $A$ is a function $g\colon A \to B$ such that $f(x)=g(x)$ for all $x\in X$ Alternatively, $g$ is an extension of $f$ to $A$ if $f$ is the restriction of $g$ to $X$

Typically, functions are not arbitrarily extended. Rather, it is usually insisted upon that extensions have certain properties. Examples include analytic continuations and meromorphic extensions.