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[parent] absolutely continuous on $[0,1]$ versus absolutely continuous on $[\varepsilon, 1]$ for every $\varepsilon >0$ (Example)
Lemma   Define $ f \colon \mathbb{R} \to \mathbb{R}$ by

% latex2html id marker 238 $\displaystyle f(x)= \begin{cases} 0 & \text{ if } x... ...isplaystyle x\sin\left( \frac{1}{x} \right) & \text{ if } x \neq 0. \end{cases}$

Then $ f$ is absolutely continuous on $ [\varepsilon, 1]$ for every $ \varepsilon >0$ but is not absolutely continuous on $ [0,1]$.

Proof. Note that $ f$ is continuous on $ [0,1]$ and differentiable on $ (0,1]$ with % latex2html id marker 257 $ \displaystyle f'(x)=\sin \left( \frac{1}{x} \right)-\frac{1}{x}\cos \left( \frac{1}{x} \right)$.

Let $ \varepsilon >0$. Then for all $ x \in [\varepsilon ,1]$:

\begin{displaymath} % latex2html id marker 263 \begin{array}{ll} \vert f'(x)\ver... ... 1 \ \ & \displaystyle =1+\frac{1}{\varepsilon} \end{array}\end{displaymath}

Since $ f$ is continuous on $ [\varepsilon, 1]$ and differentiable on $ (\varepsilon, 1)$, the mean value theorem can be applied to $ f$. Thus, for every $ x_1, x_2 \in (\varepsilon, 1)$ with $ x_1 \neq x_2$, % latex2html id marker 277 $ \displaystyle \left\vert \frac{f(x_2)-f(x_1)}{x_2-x_1} \right\vert \le 1+\frac{1}{\varepsilon}$. This yields % latex2html id marker 279 $ \displaystyle \vert f(x_2)-f(x_1)\vert \le \left( 1+\frac{1}{\varepsilon} \right) \vert x_2-x_1\vert$, which also holds when $ x_1=x_2$. Thus, $ f$ is Lipschitz on $ (\varepsilon, 1)$. It follows that $ f$ is absolutely continuous on $ [\varepsilon, 1]$.

On the other hand, it can be verified that $ f$ is not of bounded variation on $ [0,1]$ and thus cannot be absolutely continuous on $ [0,1]$. $ \qedsymbol$



"absolutely continuous on $[0,1]$ versus absolutely continuous on $[\varepsilon, 1]$ for every $\varepsilon >0$" is owned by Wkbj79.
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Cross-references: bounded variation, Lipschitz, differentiable, continuous

This is version 8 of absolutely continuous on $[0,1]$ versus absolutely continuous on $[\varepsilon, 1]$ for every $\varepsilon >0$, born on 2006-08-26, modified 2007-05-27.
Object id is 8300, canonical name is AbsolutelyContinuousOn01VersusAbsolutelyContinuousOnVarepsilon1ForEveryVarepsilon0.
Accessed 1142 times total.

Classification:
AMS MSC: 26B30 (Real functions :: Functions of several variables :: Absolutely continuous functions, functions of bounded variation)
 26A46 (Real functions :: Functions of one variable :: Absolutely continuous functions)
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something odd about this entry by silverfish on 2006-09-01 05:00:05
Have a look at the "statement of ownership" at the bottom of the entry, does it look wrong to anyone else?
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