A problem about uniform convergence

Question

$\{f_n\}$ are absolutely continuous functions on $[0,1]$, we know that if $f_n$ are uniformly convergent to a function $f$, then $f$ is continuous.

The question is: is the function $f$ absolutely continuous?

Answer 1

Recall the Weierstrass Approximation Theorem: for every continuous function $f: [0,1] \rightarrow \mathbb{R}$, there is a sequence of polynomials $P_n$ such that $P_n$ converges uniformly to $f$ on $[0,1]$.

Therefore if $P$ is any property of a function $f: [0,1] \rightarrow \mathbb{R}$ possessed by all polynomial functions, then any continuous function $f: [0,1] \rightarrow \mathbb{R}$ is a uniform limit of functions satisfying property $P$.

Try this out with $P$ being the property of absolute continuity: if

(1) Every polynomial function is absolutely continuous on $[0,1]$ and
(2) Every uniform limit of absolutely continuous functions is absolutely continuous, then
(3) Every continuous function on $[0,1]$ would be absolutely continuous.

I leave it to you to follow up on this syllogism and solve the problem.