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Convergence to the derivative

  1. Mar 31, 2008 #1
    Let

    [tex]
    f:]a,b[\to\mathbb{R}
    [/tex]

    be a continuously differentiable function, where [itex]]a,b[\subset\mathbb{R}[/itex] is some interval, and define

    [tex]
    f_n:]a,b[\to\mathbb{R},\quad
    f_n(x)=\left\{\begin{array}{ll}
    &\frac{f(x+1/n)-f(x)}{1/n},\quad x\in ]a,b-1/n[\\
    &\textrm{something continuous},\quad x\in [b-1/n, b[\\
    \end{array}\right.
    [/tex]

    Clearly we have a point wise limit [itex]f_n(x)\to f'(x)[/itex] as [itex]n\to\infty[/itex], but how common it is that this convergence is uniform? Is there some well known theorem that says that the convergence is uniform under some assumptions?
     
    Last edited: Apr 1, 2008
  2. jcsd
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