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Continuous functions on metric space, M

  1. Jan 30, 2013 #1
    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. jcsd
  3. Jan 30, 2013 #2

    Stephen Tashi

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    I suggest that you explain where you saw this statement and quote it exactly.
  4. Jan 30, 2013 #3
    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.
  5. Jan 31, 2013 #4


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