2
$\begingroup$

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?

  • 3
    If $f$ is Lebesgue integrable on $[0,x]$ for all appropriate $x$, then $F$ is absolutely continuous. See: http://en.wikipedia.org/wiki/Absolute_continuity2012-04-20
  • 0
    I guess the $M$ bound implies that $|f|$ is integrable?2012-04-21
  • 0
    @Parsa, Thanks, that was what I was looking for!2012-04-21

1 Answers 1

0

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