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Let $X,Y$ be metric spaces and $f:A\to Y$ be continuous map and $cl(A)=X$ then we know $f$ is determined on whole $X$ completely by $A$. So I want to know is now $f:X\to Y$ be necessarily be continuous ?

dragoboy
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  • I would formulate: if we want to extend $f$ to $X$ in such a way that $f:X\rightarrow Y$ is also continuous then there is at most one possibility. Then your question becomes: "is there also at least one possibility?" – drhab Feb 14 '15 at 10:03

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The tipical extension condition is $f$ to be uniformly continuous (and $Y$ complete). Otherwise, look at $f(x)=1/x$, $A=\mathbb R\setminus\{0\}$.

Jesus RS
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