Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to show the function $f:[0,1]\to \mathbb{R}$

$$f=\left\{ \begin{array}{ll} x^{3/2}\sin\left(\frac{1}{x}\right), & {x \in (0,1]} \\ 0, & x=0 \end{array} \right.$$

is absolutely continuous.

My attempt:

I broke it to functions $x^{3/2}$ and $\sin(\frac{1}{x})$. The first one is a.c. since it is increasing, for the second one I wrote the definition of absolute continuity:

$$\sin\left(\frac{1}{x_i+\delta}\right)-\sin\left(\frac{1}{x_i}\right)=2\cos\left(\frac{2x_i+\delta}{2(x_i+\delta)x_i}\right)\sin\left(\frac{-\delta}{2(x_i+\delta)x_i}\right)$$ but I don't see how it is smaller that $\epsilon\ \forall i$! Can I say for each $ϵ$ I'll find $δ=min\{ϵ,ϵ2x_i\}$ so that it converges to zero?

Another thing that I tried was using uniform integrability of ${\mbox{Diff}_\delta \ f}_{0< h\leq 1}$ but the integral results in $\Gamma$ function and imaginary number that I don't know how to handle!

share|improve this question
    
See A14. in this article mathdl.maa.org/images/cms_upload/… –  Deven Ware Dec 11 '12 at 16:57
    
I have! But I don't understand it! At $c_k \in (\frac{1}{(k+1)\pi},\frac{1}{k\pi})$ none of $f, f', f''$ are zero! take $\frac{1}{1.5\pi}$ for example –  Anita Dec 11 '12 at 17:01

1 Answer 1

About your attempt, imagine the graph of the function $\sin(1/x)$ and determine if the total variation of this function is finite.

A natural method to prove that $f$ is absolutely continuous is to calculate $f'$ and to prove that $f'$ is Lebesgue integrable. A couple of hints for this example:

  1. Suppose that $g\colon[0,1]\to\mathbb{R}$ is some function hard to integrate, but $g$ can be bounded by a simpler function $h$ (in the sense that $|g|\le h$), where $h$ is Lebesgue integrable on $[0,1]$. What can we say about $g$ in this situation?

  2. For what values of $p$ is the function $x^p$ Lebesgue integrable on $[0,1]$?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.