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What is, and what is not integrable/differentiable?

  1. May 19, 2006 #1
    I have a problem that asks me to show that a function is differentiable. Aren't all functions differentiable

    :confused:
     
  2. jcsd
  3. May 19, 2006 #2

    arildno

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    No, they are not.
    How is differentiation defined?
     
  4. May 19, 2006 #3

    George Jones

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    In fact, there exist functions that are continuous everywhere and differentiable nowhere.

    Regards,
    George
     
  5. May 19, 2006 #4

    arildno

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    And, there exist nowhere continuous functions as well. :smile:
     
  6. May 19, 2006 #5

    Hootenanny

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    Further to George's comment;
    This is continuous and is not differentiatble anywhere

    [tex]f(x) = x\sin\left(\frac{1}{x}\right) \;\; x \neq 0 \;\; f(0)=0[/tex]

    However, [itex]f(x) = |x|[/itex] is differentiatble anywhere except at [itex]x = 0[/itex]

    ~H
     
  7. May 19, 2006 #6

    benorin

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  8. May 19, 2006 #7

    matt grime

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    looks pretty differentiable everywhere but 0 to me since it is the composite and product of functions that are all differentiable away from 0.
     
  9. May 19, 2006 #8

    matt grime

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    the first of those is differentiable on a set of measure zero as the link you posted informs you.
     
  10. May 19, 2006 #9

    Hootenanny

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    Runs off to do some checking...
     
  11. May 19, 2006 #10

    matt grime

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    Why? 1/x is differentiable everywhere but 0, sin is differentiable at all points x is differentiable at all points hence xsin(1/x) is differentiable at all points except where x=0.
     
  12. May 19, 2006 #11

    Hootenanny

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    The reason I was checking was because I have this function written down as a non-differentiable function given as an example by my tutor, but it appears either she is wrong or I have copied down wrong (more likely). I've never thought to check it, till now :blushing: .

    ~H
     
  13. May 19, 2006 #12

    matt grime

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    Oh, the function is certainly not differentiable, but that is strictly different from nowhere differentiable, or not differentiable anywhere. A function is differentiable if it is differentiable at every point of its domain. So it only takes one point where it is not differentiable for the function to be 'not differentiable', yet it is differentiable everywhere except that one point.
     
  14. May 19, 2006 #13

    Hootenanny

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    Make sense, thank's matt.

    ~H
     
  15. May 19, 2006 #14

    nrqed

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    would it be possible to explain what one means by "differentiable on a set of measure zero"? Or to point to a website explaining this?
    Thanks!
     
  16. May 19, 2006 #15

    Curious3141

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    This says something about it. http://en.wikipedia.org/wiki/Measure_zero

    My understanding (not great) from that is that a set of measure zero is necessarily a null set (which is important here, there are no points at which the function is differentiable). The converse is not necessarily true (which doesn't seem important here).
     
  17. May 19, 2006 #16

    arildno

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    Roughly spoken, the measure of a set says how big it is.
    In the plane, we might say that the measure of a set of points is the area of the region consisting of those points.

    Consider now a line lying in the plane.
    What is the area of that line?
    Clearly, a line should have zero area, and this corresponds to saying that the set of points constituting the line is a set of measure zero.
     
  18. May 19, 2006 #17

    George Jones

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    I think you've got it the wrong way round.

    If A is the null set, then necessarily A has measure zero, but, if A has measure zero, then A is not necessarily the null set. For example, if A is the set of all rational number, then, as a subset of the standard measure space of real numbers, A has measure zero.

    Regards,
    George
     
  19. May 19, 2006 #18

    Curious3141

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    Not according to the Wiki article I linked. Is Wiki wrong? (I honestly don't know, I was just lifting from there).
     
  20. May 19, 2006 #19

    George Jones

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    Okay, in this context, Wiki defines a null set to be a set of measure zero. Consequently, in my example, the set of all rational numbers is a null set.

    But from

    I took it that you meant the empty set, which is also sometimes called the null set.

    Measure zero sets are not necessarily empty.

    Regards,
    George
     
  21. May 19, 2006 #20

    nrqed

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    Thanks, I had some idea of measure in general but I was wondering about "*differentiable* on a s et of measure zero". This means that the functions *is* differentiable on a (possibly infinite) number of points but that those points form a set of measure zero. Ok, but I guess I wonder how that looks like or how the proof is done.

    Thanks for the help!
     
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