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Question about defn. of derivative

  1. Apr 18, 2010 #1
    Let [itex]\{h_n\}[/itex] be ANY sequence of real numbers such that [itex]h_n \neq 0[/tex] and [itex]h_n \to 0[/itex]. If [itex]f'(x)[/itex] exists, do we have

    [tex]
    f'(x) = \lim_{n\to \infty} f_n(x),
    [/tex]

    where
    [tex]
    f_n(x) = \frac{1}{h_n} (f(x+h_n) - f(x))
    [/tex]

    ????

    This seems to express the derivative as the pointwise limit of a sequence of functions...right? Do we know, in addition, that the convergence is uniform?
     
  2. jcsd
  3. Apr 18, 2010 #2

    mathman

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    Uniform in what sense? The definition does not require uniformity in x.
     
  4. Apr 18, 2010 #3
    Ok, I'm not sure :) I didn't really think before writing out that question. But am I right on all other counts?
     
  5. Apr 19, 2010 #4

    mathman

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    You are right, but it is more cumbersome than the usual approach where:

    f'(x)=lim(h->0) (f(x+h) - f(x))/h
     
  6. Apr 20, 2010 #5

    HallsofIvy

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    [itex] lim_{x\to a} f(x)= L[/itex] if and only if [itex]\lim_{n\to \infty} f(a_n)= L[/itex] for every sequence [itex]\{a_n\}[/itex] that converges to a. The two formulations are equivalent.
     
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