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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.2012-04-19
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    @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.2012-04-19
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    @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 !2012-04-19
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    @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" ?2012-04-19
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    @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?2012-04-19
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    @MartinWanvik: I am trying to find an example that has not been answered by the comments of Robert and Alex above - though at the moment it seems I might have mixed things up in my head because I cannot find one. I'll edit the post in case I find one !2012-04-19

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