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Pointwise convergence

  1. Apr 9, 2006 #1
    If a sequence of functions f_n converges pointwise to a bounded function f, does f have the intermediate value property? If not, are there some conditions that will make it so?
  2. jcsd
  3. Apr 9, 2006 #2
    No. Consider f_n = f for all f and f(x) = 0 for x<1 and 1 for x geq 1.

    The intermediate value property holds for continuous functions. But even then pointwise convergence is not enough. Consider f_n=x^n on [0,1]. The limiting function is f(x)=0 for x in [0,1) and f(1)=1.

    However, the limit of a sequence of continuous functions will be continuous if you have uniform convergence.
  4. Apr 9, 2006 #3

    matt grime

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    As stated this is trivially false. Surely you mean f_n to be continuous at the very least.
  5. Apr 9, 2006 #4
    I don't have uniform convergence. But suppose (f_n) is an increasing sequence of continuous functions on R. Suppose (f_n) converges pointwise to f, then {x in R : f(x)>a} must be open, and I can show this if I had the intermediate value property.
  6. Apr 9, 2006 #5
    Actually I don't think this is true without uniform convergence. I can think of functions that satisfy the intermediate value property and yet {x in R : f(x)>a} is not open, even closed. I'll repost in the homework section.
    Last edited: Apr 9, 2006
  7. Apr 10, 2006 #6


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    Let fn(x)= 0 is x<0, nx if 0<= x<= 1/n, 1 if x> 1/n.

    Each such f is continuous but the sequence converges pointwise to
    f(x)= 0 if x<= 0, 1 if x> 0 which is not continuous and does not satisfy the intermediate value property.
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