Continuous functions on metric space, M

roman93

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!

Stephen Tashi

Quote by roman93

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

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

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

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roman93	Continuous functions on metric space, M 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!

micromass Mentor Blog Entries: 8	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 Recognitions: Gold Member Science Advisor Staff Emeritus	Continuous functions on metric space, M Since f(p)= p is continuous, one thing that tells you is that M itself is bounded!