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Thread: 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 {$\displaystyle f_{n}$}$\displaystyle _{n \in N}$ 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 {$\displaystyle f_{n}$}$\displaystyle _n \in N$ such that:

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

    d.) {|$\displaystyle f_{n}$|}$\displaystyle _{n \in N}$ converges pointwise but {$\displaystyle f_{n}$}$\displaystyle _{n \in N}$ 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 {$\displaystyle f_{n}$}$\displaystyle _{n \in N}$ 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 {$\displaystyle f_{n}$}$\displaystyle _n \in N$ such that:

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

    d.) {|$\displaystyle f_{n}$|}$\displaystyle _{n \in N}$ converges pointwise but {$\displaystyle f_{n}$}$\displaystyle _{n \in N}$ 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 $\displaystyle (-1)^n$
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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 $\displaystyle \in$ [-1,1];
    otherwise, h(x-2) if x > 0 or h(x+2) if x < 0

    f(x) = $\displaystyle \sum (\frac{3}{4})^n$ h($\displaystyle 4^nx$) with n going from 0 to $\displaystyle \infty$
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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 $\displaystyle \in$ [-1,1];
    otherwise, h(x-2) if x > 0 or h(x+2) if x < 0

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