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Continuous => limited in region

  1. Dec 16, 2006 #1
    1. The problem statement, all variables and given/known data

    I've to show that if f:R->R is continuous in x' then f is limited in a suitable environment of x'.

    2. The attempt at a solution

    My lecturer said we should use the following inequality

    |f(a)|=<|f(a)-f(y)|+|f(y)|

    But how should I go on, I know I have to show something like this:

    It exists d>0: it exist c element of [x'-d, x'+d]=I ==> |f(x)|=<|f(c)| for every x in I.

    But how should I go on?
     
  2. jcsd
  3. Dec 16, 2006 #2

    StatusX

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    Always include relevant definitions. "limited in a suitable environment" is not standard terminology, and I don't know what it means. It sounds like "bounded in some neighborhood", but this would follow imediately from the epsilon delta definition of continuity.
     
  4. Dec 16, 2006 #3
    I thought that, too. I mean that it follows imidiately, but our lecturer said that we've to take |f(a)|=<|f(a)-f(y)|+|f(y)| to show that.

    With limited in a suitable environment it is ment what you said (bounded in some neightboorhood).
     
  5. Dec 16, 2006 #4

    StatusX

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    Right, so pick any e>0, then a d>0 so that |y-a|<d => |f(y)-f(a)|<e, and then use that inequality to show there is an M with |f(y)|<M, all y in some neighborhood of a.
     
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