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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!

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2 Answers

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.

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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...

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