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Differentiable Function on an interval

  1. Dec 4, 2011 #1
    Let f:[a,b][itex]\rightarrow[/itex]R be continuous on [a,b] and differentiable in (a,b). Show that if lim f'(x)=A as x goes to a then f'(a) exist and equals A.


    So I was thinking this has to do either with the mean value theorem or Darboux's Theorem.

    I have that
    f(b)-f(a)=f'(c)(b-a) by the mean value theorem.

    From here I stuck on how to get the x into the equation.
    Would I say that let x=c.

    Then we have
    f'(x)=[itex]\frac{f(b)-f(a)}{b-a}[/itex]=A


    If so how would I work in the f'(a)?
     
  2. jcsd
  3. Jun 27, 2012 #2
    It really comes down to the definitions of continuity and differentiability. They are asking you to show that:
    if f(x) is continuous for all x [itex]\in[/itex] [a,b]
    and if f(x) is differentiable for all x [itex]\in[/itex] (a,b)
    and if lim f'(x) as x->a exists
    then f'(a) exists
    f(x) is differentiable in the interval (a,b) and the one-sided limit as x->a of f'(x) exists, therefore f'(a) exists. If f'(a) exists, then f'(x) is differentiable at x=a. Differentiability implies continuity, and so f'(x) would also be continuous at x=a. If f'(x) is continuous at x=a, then you can say by the definition of continuity that:
    [itex]\stackrel{lim}{x→a} f'(x) = f'(a) = A[/itex]
     
    Last edited: Jun 27, 2012
  4. Jun 27, 2012 #3

    HallsofIvy

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    "If f'(a) exists, then f'(x) is differentiable at x= a" That is, that f is twice differentiable at x= a? Why is that true?

    It looks to me like the statement in the first post is not even true. Let f(x)= |x|. Then f is continuous for all x and so on [0, 1]. Further, f is differentiable in (0, 1)- its derivative is just 1 for all x in (0, 1). The limit of f'(x), as x goes to 0, is, of course, 1 but f is NOT differentable at x= 0.
     
  5. Jun 27, 2012 #4
    Sorry, you are correct HallsofIvy. I don't know how I missed that. Well, at least the first part is correct, f'(a) exists. I guess the question then is does continuity and differentiability of f imply continuity of df/dx? I know that f(x) being differentiable alone doesn't necessarily prove it, but if f(x) is also continuous then I see no reason why df/dx is not continuous. I have googled it extensively with not much luck.
     
  6. Jun 27, 2012 #5
    What about this...
    [itex]f'(x) = \stackrel{lim}{s→x}\frac{f(s)-f(x)}{s-x}[/itex]

    [itex]\stackrel{lim}{x→a}f'(x)[/itex]
    [itex]= \stackrel{lim}{(x→a)} \stackrel{lim}{(s→x)}\frac{f(s)-f(x)}{s-x}[/itex]
    [itex]= \stackrel{lim}{s→a}\frac{f(s)-f(a)}{s-a} = f'(a)[/itex]

    [itex]\stackrel{lim}{x→a}f'(x) = f'(a) = A[/itex]
     
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