
#1
Jan3013, 08:44 AM

P: 14

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! 



#2
Jan3013, 11:33 AM

Sci Advisor
P: 3,170





#3
Jan3013, 11:38 AM

Mentor
P: 16,542

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. 



#4
Jan3113, 09:11 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879

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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