Given $C^1([a, b])$ functions $f\_n$ that converge to a continuous real-valued function $f\_n \to f$ on a closed interval $[a, b] \subset \mathbb R$, suppose
$$
\int\_a^b |f\_n'(x)|^{1 + \epsilon} dx < M
$$
for all $n > 0$ and fixed $M < \infty$. For a small $\epsilon$, does this implies


1. $f$ is absolutely continuous
2. $f$ is Lipschitz


Can we tell more about this function?


I have $\epsilon = 0$ does not hold for Cantor function and some approximation $f\_n$.



Does that exponent of (absolute value of derivative) is constrained implies Lipschitz continuity?
Q1: Yes. Since $f'\_n$ is bounded in $L^p$, a suitable subsequence will converge $f\_n'\to g$ weakly. Then also
$$
f(x) =\lim f\_n(x) = \lim \left( f\_n(a)+\int\_a^x f'\_n(t)\, dt \right) = f(a) + \int\_a^x g(t)\, dt ,
$$
so $f$ is absolutely continuous (with derivative in $L^p$).


Q2: No. For example $f(x)=|x|^{\alpha}$ has a derivative $f'\in L^p$ for $p<1/(1-\alpha)$, so can easily be approximated by functions with bounded $L^p$ derivatives.


