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Simple Question regarding C[0,1]

  1. Apr 14, 2005 #1
    So if D={f is an element of C[0,1];f(1) does not equal 1}

    and C[0,1] is the set of complex valued continuous functions on the interval [0,1], is there a function f such that f approaches 1 and f(1) does not equal 1?

    I feel like there has to be one but I am unable to construct one since if the lim as x approaches 1 does not equal f(1)=1, then f wouldn't be continuous right?

    I'm trying to show that D doesn't contain all of its limit points since that would be all that is required to show D is not closed.

    Help with finding this function if there is one please.
  2. jcsd
  3. Apr 14, 2005 #2
    I don't understand what you are looking for. If you're trying to show that D is not closed by finding a limit point outside of D, then all you have to do is find a sequence of functions in D that converges to a function not in D.

    For example, take f_n(x) = n / (n + 1). Then every f_n is in D, but the sequence converges to f(x) = 1, which is not in D.
  4. Apr 14, 2005 #3
    Yeah, you basically answered my question
  5. Apr 14, 2005 #4
    But then why were you asking about trying to find a function such that f(1) != 1 but f(x) -> 1 as x -> 1?
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