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Math Help - Lower and upper derivaties of function at point

  1. #1
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    Lower and upper derivaties of function at point

    Hi guys,
    I'm looking for a little help with computing the lower and upper derivatives of my function given below, at a point x\in (a, b).

    f(x) = \sum_{k=1}^{\infty}\left [ {length(  (c_{k}, d_{k})\cap (-\infty, x)    )} \right ]

    (where  \left \{ (c_{k}, d_{k})_{k=1}^{\infty} \right \} is a countable collection of open intervals in (a, b) and \sum_{k=1}^{\infty}(d_{k} - c_{k}) < \infty)

    The lower and upper derivatives of a real-value function are defined as such:
    Uf(x) = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{f(x+t) - f(x)}{t}}\right]

    Lf(x) = \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{f(x+t) - f(x)}{t}}\right]

    This is what I have, could someone tell me if this is wrong? I have some doubts.

    If f(x+t)= \sum_{k=1}^{\infty}  \left[ min(d_{k}, x+t) - c_{k}\right ]

    Uf(x) = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{  \sum_{k=1}^{\infty}  \left[ (min(d_{k}, x+t) - c_{k}) - (min(d_{k}, x) - c_{k})  \right ]}{t}}\right]

    = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{  \sum_{k=1}^{\infty}  \left[ min(d_{k}, x+t) - min(d_{k}, x)  \right ]}{t}}\right] = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{  \sum_{k=1}^{\infty}  \left \{ 0 \leqslant y \leqslant t \right \}          }{t}}\right] = \infty


    Similarly,

    Lf(x)= \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{  \sum_{k=1}^{\infty}  \left[ min(d_{k}, x+t) - min(d_{k}, x)  \right ]}{t}}\right] = \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{  \sum_{k=1}^{\infty}  \left \{ 0 \leqslant y \leqslant t \right \}          }{t}}\right] = 0
    Last edited by director; February 4th 2014 at 05:59 PM.
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