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Math Help - Need examples for functions that fit the following

  1. #1
    Member thaopanda's Avatar
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    Need examples for functions that fit the following

    Give an example, if possible, of a sequence { f_{n}} _{n \in N} of discontinuous functions on R which converges uniformly to:

    a.) a discontinuous function on R

    b.) a continuous function on R

    Give an example of a sequence of functions { f_{n}} _n \in N such that:

    c.) f_{n} is continuous and not differentiable but the sequence converges pointwise to a differentiable function

    d.) {| f_{n}|} _{n \in N} converges pointwise but { f_{n}} _{n \in N} does not
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  2. #2
    Member Focus's Avatar
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    Quote Originally Posted by thaopanda View Post
    Give an example, if possible, of a sequence { f_{n}} _{n \in N} of discontinuous functions on R which converges uniformly to:

    a.) a discontinuous function on R

    b.) a continuous function on R

    Give an example of a sequence of functions { f_{n}} _n \in N such that:

    c.) f_{n} is continuous and not differentiable but the sequence converges pointwise to a differentiable function

    d.) {| f_{n}|} _{n \in N} converges pointwise but { f_{n}} _{n \in N} does not
    a) Take the constant sequence of a discontinuous function
    b) Take the sequence f(x)=0 if x<0, f(x)=1/n otherwise
    c) Do you mean nowhere differentiable or only at one (or more) point?
    d) Take (-1)^n
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  3. #3
    Member thaopanda's Avatar
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    Nowhere differentiable

    That was all the problem said, so I take it that it should be nowhere differentiable. The only thing I can think of that is nowhere differentiable is:

    g(x) = |x| on [-1,1]
    h(x) = g(x) if x \in [-1,1];
    otherwise, h(x-2) if x > 0 or h(x+2) if x < 0

    f(x) = \sum (\frac{3}{4})^n h( 4^nx) with n going from 0 to \infty
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    That was all the problem said, so I take it that it should be nowhere differentiable. The only thing I can think of that is nowhere differentiable is:

    g(x) = |x| on [-1,1]
    h(x) = g(x) if x \in [-1,1];
    otherwise, h(x-2) if x > 0 or h(x+2) if x < 0

    f(x) = \sum (\frac{3}{4})^n h( 4^nx) with n going from 0 to \infty
    So maybe take F_k(x)=\frac{f(x)}{k}. It's a pretty stupid example, but it does converge to 0. I believe the convergence is uniform, but I'm not sure.
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