Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Here's a question I came up randomly just now:

If a sequence of absolutely continuous functions $\{\phi_{n}\}$ converges uniformly to a function $f$, does it imply that $f$ is also absolutely continuous?

I am having trouble judging if this is true. If the statement is not true, can anybody help me find a counter-example?

Thank you!

share|cite|improve this question
up vote 4 down vote accepted

No, it does not. Even the set of smooth functions, or the set of polynomials is dense in $C^0$ with respect to uniform convergence, on an interval, say (which is just convergence in the supremums norm). That is, to each continuous function $f$ you will find a sequence $f_k$ of smooth functions converging to $f$ uniformly. This sequence is, in particular, a sequence of AC functions.

share|cite|improve this answer

I wonder if Cantor's function is a uniform limit of absolutely continuos functions, and yet it is not AC. More generally, I think that a uniform convergence cannot pass to the first derivative: I can't see how the limit function can have a summable derivative under the mere assumption that its approximants do...

share|cite|improve this answer
It's continuous, so it is the uniform limit of smooth functions. – user20266 Apr 22 '12 at 15:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.