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Suppose we have a function $f:X \rightarrow Y$. Now, consider the function $g:X'\rightarrow Y$ where $X'\subset X$.

I'd like to say the $g$ is a "subset" of $f$ ; is there a correct term for describing $g$ w.r.t $f$?

NB: Also, $g(x) = f(x)$ for $x \in X'$

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Provided $g(x)=f(x)$ for all $x \in X'$, we say that $g$ is the restriction of $f$ to $X'$, and that $f$ is an extension of $g$ to $X$.

If $g$ is the restriction of $f$ to $X'$ we can write $g=f|_{X'}$.

Restrictions are unique (hence 'the' restriction), extensions are not (hence 'an' extension).

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    Just checking (just in case the term was for functions where the domains were related and codomains were the same). Thanks, again!2012-09-14
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What you are referring to is the restriction of $f$ to $X'$ — assuming that you define $g(x) =f(x)$ your all $x \in X'$, that is.