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Functional analysis

  1. Jan 27, 2008 #1
    Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is:

    Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2
    Let A={g Є C[0,1]: d(g,f) > 3}. Prove that A is an open set
    Let B={g Є C[0,1]: 1 ≤ d(g,f) ≤ 3}. Prove that B is a closed set

    I'm new to all of this and just don't know what to do even with the f(x)=x²+2 part so if anyone out there can shed some light, I'd be really grateful!

  2. jcsd
  3. Jan 27, 2008 #2


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    Homework Helper

    You're trying to prove that sets are open and closed, not functions. I suggest you sketch the graph of f. Here, d(f,g) represents the maximum distance between two continuous functions f and g. So play around with the sketch, and then try to prove your observations.
  4. Jan 28, 2008 #3
    Thank you for your advice, I'll try that and see how I get on

    Thanks again
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