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This is a very short question, I hope it is not too broad, if so I shall try and make it more specific. I would like to start as it stands below, though, because it really points down the essence of the question:

What do people mean when they write that a map is extended by continuity ?

Thanks !

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Generally I see this when you have a continuous function $f$ defined on a subset $Y$ of a metric space $X$ (or sufficiently nice topological space) and want a continuous function $g$ on $X$ such that $f(x)=g(x)$ for all $x\in Y$. One then says $g$ extends $f$. It is a theorem that $g$ exists if $Y$ is closed. – Alex Becker Apr 19 '12 at 22:56
@AlexBecker: that's extending continuously, but not extending by continuity. I'd say $f$ extends by continuity if $Y$ is a dense subset of $X$ and for each $x \in X$, $\displaystyle \lim_{y \to x, y \in Y} f(y)$ exists. – Robert Israel Apr 19 '12 at 23:07
@Alex: Hm .. ok, though sometimes a map is "extended by continuity" to the dual space - and this is not a subspace. I realize this concept might depend on the context, and your comment makes sense of some of the instances I hace struggled with, so thanks ! – harlekin Apr 19 '12 at 23:10
@RobertIsrael: many thanks for the helpful comment! May I ask, what is meant if people define a map on the dual space by "extension by continuity" ? – harlekin Apr 19 '12 at 23:12
@harlekin: it would be helpful if you could provide more context (such as a relevant quote from a book). What kind of map is being extended? From where? – Martin Wanvik Apr 19 '12 at 23:21

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