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Thread: Clarification about Differentiation

  1. #1
    Super Member sakonpure6's Avatar
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    Clarification about Differentiation

    My professor said in class:
    The derivative of f at x=a is f'(a) = [f(a+h) - f(a) ] /h (and said that this Limit does not need to exist for the derivative to exist). (Then he said) If f'(a) exists, then f is differentiable at x=a
    Can someone please give me an example of a case where the limit does not need to exist at x=a for the function to have a derivative at x=a ?

    If f'(a) exists, does that mean f is continuous on x=a? ( I think yes) But this is not the other way around?

    Thank you in advance.
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  2. #2
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    Re: Clarification about Differentiation

    I am NOT going to assume that you are quoting your professor correctly since he/she is not here to define him/herself! No, [f(x+ h)- f(x)]/h is NOT the derivative- it is the "difference quotient" and the derivative is the limit of that as h goes to 0. That limit must exist in order that the function be "differentiable" there.

    Now, one of the basic properties of limits is that if both \lim_{x\to a} f(x) and \lim_{x\to a} g(x) exist, then \lim_{x\to a}\frac{f(x)}{g(x)} exists. However, for the difference quotient, since the denominator is h, the denominator will always go to 0- but the limit may exist anyway. Perhaps that is what your professor was trying to tell you.
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  3. #3
    Senior Member zzephod's Avatar
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    Re: Clarification about Differentiation

    Quote Originally Posted by sakonpure6 View Post
    My professor said in class:


    Can someone please give me an example of a case where the limit does not need to exist at x=a for the function to have a derivative at x=a ?

    If f'(a) exists, does that mean f is continuous on x=a? ( I think yes) But this is not the other way around?

    Thank you in advance.
    It depends on what you mean by derivative, in the distributional sense the derivative of a step is a dirac delta, but here the derivative of distributions is not defined as a limit but by the identity:

    $$\int_{-\infty}^{\infty}f'g =-\int_{-\infty}^{\infty}fg'$$

    where $f$ is our distribution and $g$ a tempered distribution (infinitely differentiable and goes to zero (with all its derivatives) as $x$ goes to $\pm \infty$) with the usual definition of derivative for tempered distributions.

    .
    Last edited by zzephod; Sep 24th 2014 at 01:21 PM.
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  4. #4
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    Re: Clarification about Differentiation

    I think either you misheard or your teacher misspoke.

    $\displaystyle \lim_{h \rightarrow 0}\dfrac{f(a + h) - f(a)}{h}\ may\ or\ may\ not\ exist,\ but\ \lim_{h \rightarrow 0}\dfrac{f(a + h) - f(a)}{h}\ exists \iff f(x)\ is\ differentiable\ at\ a.$

    And differentiability does imply continuity, but continuity does not imply differentiability.
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