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If a function f is differentiable at a point x = c of its domain, then

  1. Apr 9, 2009 #1
    If a function f is differentiable at a point x = c of its domain, then must it also be differentiable in some neighborhood of x = c?
     
  2. jcsd
  3. Apr 9, 2009 #2
    Re: Derivatives

    In some neighborhood yes. Proof of derivation relies on this (if I'm correct). If 2.9999 is differentiable and 3.0000 is not, there is still the 2.99999 or 2.999999 between the 2 values, so there is some local surrounding domain you could take into account.
     
  4. Apr 10, 2009 #3
    Re: Derivatives

    No this is not true. Consider the function:
    [tex]f(x) = \left\{\begin{array}{lr}x^2&x\in\mathbb{Q}\\-x^2&x\not\in\mathbb{Q}\end{array}[/tex]
    This function is only differentiable at the point x=0.
     
  5. Apr 10, 2009 #4
    Re: Derivatives

    Good catch, slider142! Okay, for fun let's emend the question a bit...

    Given an function that is continuous in some neighborhood of c and differentiable at c, must it also be differentiable in some neighborhood of c?
     
  6. Apr 12, 2009 #5
    Re: Derivatives

    Ahh, good question;p

    Isn't slider142's function
    continuous at x=0? Isn't it necessary for a function to be continuous at a point c for it to be derivable at c?

    And what does it mean to be continuous at a point? Doesn't continuity necessarily involve a neighbourhood of a point? So I think first we must ask, if a function is continuous at a point c, then is it continuous in a neighbourhood of that point?
     
  7. Apr 12, 2009 #6
    Re: Derivatives

    a function f is continuous at a if
    [tex]\lim_{x \to a}f(x) = f(a)[/tex]

    Continuity at a point only means that nearby points approach a, not that it's continuous near a.
     
  8. Apr 12, 2009 #7
    Re: Derivatives

    I don't think so. Take any continuous function f differentiable at c and any continuous, nowhere differentiable function g bounded in some neighbourhood of 0 (such as the Weierstrass function). Then f(x) + (x-c)*g(x-c) should be continuous, differentiable at c, but not differentiable on any neighbourhood of c.
     
  9. Apr 12, 2009 #8
    Re: Derivatives

    -nods at qntty-

    but continuity at a point does require the existence of the function in a neighbourhood of the point. Slider's function is continuous at x=0 but at no other point (I think) :p
     
  10. Apr 12, 2009 #9
    Re: Derivatives

    Indeed. The common example of a function continuous at a single point but defined for all x in R is
    [tex]f(x)=\left\{\begin{array}{lr}x,&x\in\mathbb{Q}\\0,&x\not\in\mathbb{Q}\end{array}[/tex]
    or you can replace the 0 function by -x, etc.
    From the definition of derivative it is easy to show directly that if the function is differentiable at a point, it is also continuous there. The converse is not true, as exemplified by this function and the Weierstrauss function.
     
    Last edited: Apr 13, 2009
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