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Differentiability of the mean value

  1. Aug 4, 2007 #1
    So if a function

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

    is differentiable, then then for each [itex]x\in [a,b][/itex] there exists [itex]\xi_x \in [a,x][/itex] so that

    [tex]
    f'(\xi_x) = \frac{f(x)-f(a)}{x-a}
    [/tex]

    Sometimes there may be several possible choices for [itex]\xi_x[/itex]. My question is, that if the mapping [itex]x\mapsto \xi_x[/itex] is chosen so that it is continuous, is it always also differentiable? In other words, does the limit

    [tex]
    \lim_{h\to 0}\frac{\xi_{x+h}-\xi_x}{h}
    [/itex]

    exist?
     
  2. jcsd
  3. Aug 4, 2007 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    what you wrote is correct when: f'(e)(x-a)=f(x)-f(a), cause of the choice of x=a.
    for your question you mean the function at the points ksi_x+h and at ksi_x.

    well this is ofcourse correct when f is differentaibale continuous.
    btw, it's enough to assume that it's differentiable in (a,b) and continuous in [a,b].
     
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