Continuous functions on metric space, M


by roman93
Tags: continuous, functions, metric, space
roman93
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#1
Jan30-13, 08:44 AM
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If every continuous function on M is bounded, what does this mean?

I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!
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Stephen Tashi
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Jan30-13, 11:33 AM
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Quote Quote by roman93 View Post
is it a mapping from M -> M or some other mapping?
I suggest that you explain where you saw this statement and quote it exactly.
micromass
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Jan30-13, 11:38 AM
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A map [itex]f:X\rightarrow M[/itex] where X is a set and M is a metric space, is called bounded if the image f(X) is bounded. This means that there is an [itex]x\in X[/itex] and an [itex]\varepsilon>0[/itex] such that [itex]f(X)\subseteq B(f(x),\varepsilon)[/itex].

This is what I would call bounded. But you will need to specify the context.

HallsofIvy
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Jan31-13, 09:11 AM
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Continuous functions on metric space, M


Since f(p)= p is continuous, one thing that tells you is that M itself is bounded!


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