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Example of an absolutely continuous function on [a,b] whose derivative is unbounded.

  1. Oct 28, 2011 #1
    What is an example of an absolutely continuous function on [a,b] whose derivative is unbounded?

    I know that the function f: [-1,1] defined by f(x) = x^2sin(1/x^2) for x ≠ 0, f(0) = 0 is continuous and its derivative f'(x) = 2xsin(1/x^2)-2/xcos(1/x^2) for x ≠ 0, f'(0) = 0 is unbounded on [-1,1]. But this function isn't absolutely continuous...

    Any help would be much appreciated : )
     
  2. jcsd
  3. Oct 28, 2011 #2
    Re: Example of an absolutely continuous function on [a,b] whose derivative is unbound

    We can construct another example along the same lines. Consider the following:

    [tex]f(x)=\begin{cases} x^2 \sin (|x|^{-3/2}) & \text{if }x \neq 0 \\ 0 & \text{if } x=0 \end{cases}[/tex]

    Computing the derivative yields:

    [tex]f'(x) = \begin{cases} 2x \sin (|x|^{-3/2}) - \frac{3}{2} \operatorname{sgn}(x) |x|^{-1/2} \cos(|x|^{-3/2}) & \text{if } x \neq 0 \\ 0 & \text{if } x=0 \end{cases}[/tex]

    Since the derivative is integrable on [-1, 1], f is absolutely continuous on that interval.
     
  4. Oct 30, 2011 #3
    Re: Example of an absolutely continuous function on [a,b] whose derivative is unbound

    Thank you for your help. One question: How I do know that the derivative of that function is integrable? If f' were bounded, then the fact that it is only discontinuous at x = 0 would make f' Reimann integrable, and Reimann integrability implies Lebesgue integrability for bounded functions. But in this case, f' is not bounded, so how do I show that it is Lebesgue integrable?
     
  5. Oct 30, 2011 #4
    Re: Example of an absolutely continuous function on [a,b] whose derivative is unbound

    f' is obviously measurable, so you only have to show that the integral of the absolute value is finite. But we have:

    [tex]\begin{align}\int_{-1}^{1} |f'(x)| \ dx & \leq \int_{-1}^{1} 2|x| |\sin(|x|^{-3/2})| + \frac{3}{2}|x|^{-1/2} |\cos(|x|^{-3/2})| \ dx \\ &\leq \int_{-1}^{1} 2 + \frac{3}{2} |x|^{-1/2} \ dx \\ & = 4 + \frac{3}{2} \int_{-1}^{1} |x|^{-1/2}\ dx \\ & = 4 + 3 \int_{0}^{1} x^{-1/2} dx \\ & = 4 + 6 \sqrt{x} \Big\vert_{0}^{1}\\ & = 10 < \infty \end{align}[/tex]
     
  6. Oct 30, 2011 #5
    Re: Example of an absolutely continuous function on [a,b] whose derivative is unbound

    Aha. Thank you! I knew that if a bounded function is Reimann integrable then the Reimann integral and the Lebesgue integral of the function are equal. However, I always forget that this can be extended to unbounded functions as long as the Reimann integral is finite.

    Thanks again!
     
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