# If the Lebesgue integral $\int_0^x f(y) \; dy$ is bounded, must it be continuous?

This question is out of my curiosity, I have finished my calculus course years ago and unfortunately all the knowledge became rusty, right now I cannot deal even with this simple-looking question.

Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function such that $F(x) = \int_0^x f(y)\ \mathrm{d}y$ exists and $\|F\|_\infty = M$ is bounded ($\int$ denotes the Lebesgue's integral). Must $F$ be continuous?

If not, could someone sketch the counter-example?

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If $f$ is Lebesgue integrable on $[0,x]$ for all appropriate $x$, then $F$ is absolutely continuous. See: en.wikipedia.org/wiki/Absolute_continuity – Parsa Apr 20 '12 at 23:12
I guess the $M$ bound implies that $|f|$ is integrable? – copper.hat Apr 21 '12 at 2:39
@Parsa, Thanks, that was what I was looking for! – dtldarek Apr 21 '12 at 8:21

## 1 Answer

Parsa's comment answers the question:

If $f$ is Lebesgue integrable on $[0,x]$ for all appropriate $x$, then $F$ is absolutely continuous. See Absolute Continuity

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