TITLE: Limit of absolutely continuous functions
QUESTION [5 upvotes]: Is $AC[a,b]$ a closed subspace of $C[a,b]$. If so how does one go about proving this, and if not why? (The proof that it is a subspace is clear)
Any help is appreciated.

REPLY [3 votes]: Take $f(x) = \begin{cases} x \sin \frac{1}{x} & x \neq 0 \\ 0 & x=0 \end{cases}$. Then let $f_n(x) = \begin{cases} x \sin \frac{1}{x} & |x| > \frac{1}{n}  \\ x f(\frac{1}{n}) & |x| \leq \frac{1}{n} \end{cases}$. Then $f_n \to f$ uniformly, and $f_n$ is absolutely continuous (since Lipschitz), but $f$ is not absolutely continuous.