Is there a function that is uniformly continuous function but not absolutely continuous.

My answer is $f(x)=x^{2}, \forall x\in R$

Is this right?

Are there any other?

  • 2
    $\begingroup$ Your example is not uniformly continuous. You cannot guarantee that for all real numbers a and b, there exists a delta such that |f(a) - f(b)| = |a^2 - b^2| < delta whenever |a-b| is small. You can make it uniformly continuous by restricting its domain to a bounded, closed subset of R (a closed interval [a, b]) $\endgroup$
    – Bart W
    May 27, 2016 at 11:19
  • $\begingroup$ @BartW , Is there a simpler example that is uniformly continuous but not absolutely continuous? $\endgroup$
    – MrDi
    May 27, 2016 at 11:27

1 Answer 1


The Cantor function is an example.

  • $\begingroup$ Is there a simpler example? $\endgroup$
    – MrDi
    May 27, 2016 at 11:34
  • $\begingroup$ @MrDi I would suppose that the Cantor function is the simplest function which is not absolutely continuous. If you have a simpler function which is continuous, but not absolutely continuous, you can restrict it to a compact interval. $\endgroup$
    – gerw
    May 27, 2016 at 17:56
  • $\begingroup$ I think your answer is wrong cantor functions isn't uniformally continous $\endgroup$
    – Az264
    Dec 31, 2017 at 7:08
  • 1
    $\begingroup$ @Az264 It actually is. A continuous function on a compact set is uniformly continuous. $\endgroup$
    – suncup224
    Oct 28, 2018 at 2:17

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .