TITLE: Does uniform convergence preserves absolute continuity?
QUESTION [6 upvotes]: 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!

REPLY [3 votes]: 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...