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Let $I$ be some bounded interval(if it's closed,then the result is trivial).

The question is if $I$ is only bounded (for example open bounded),can we claim that absolute continuous function $f:I\to \Bbb{R}$ is integrable/bounded on that interval?

My attempt,we may try to divide the interval into two part then for example let $I = (a,a_1)\cup[a_1,a_2]\cup(a_2,b)$

we may choose $|a_1-a| = |a_2-b|\le \delta$ for some small delta,then $f(x)$ inside this two side part region is bounded by $\max(f(a_1)+1,f(a_2)+1)$ (by uniformly continuous of $f$ on $I$) so combine these three region it's bounded.Is my idea correct?

yi li
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1 Answers1

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Absolutely continuous functions are uniformly continuous, and therefore they are bounded on any bounded interval.