# Does uniform convergence preserves absolute continuity?

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!

-

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.