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Let $f_n$ be a sequence of continuous, differentiable a.e. functions with

1. $f_n \to f$ uniformly for some continuous $f$.
2. $f'_n - g \to 0$ in $L^\infty$ for some measurable $g$.

where we assume that $f'_n - g$ is in $L^\infty$ for all large enough $n$.

Is it true that $f$ is differentiable a.e. with $f' = g$ a.e.?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.

Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with

1. $f_n \to f$ uniformly for some continuous $f$.
2. $f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,

where we assume that $f'_n - g$ is in $L^\infty$ for all large enough $n$.

Is it true that $f$ is differentiable a.e. with $f' = g$ a.e.?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.

Let $f_n$ be a sequence of continuous, differentiable a.e. functions with $f_n \to f$ uniformly and $f'_n - g \to 0$ in $L^\infty$ for some measurable functions $f, g$, where we assume that $f'_n - g$ is in $L^\infty$ for all large enough $n$.

Is it true that $f$ is differentiable a.e. with $f' = g$ a.e.?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.

Let $f_n$ be a sequence of continuous, differentiable a.e. functions with

1. $f_n \to f$ uniformly for some continuous $f$.
2. $f'_n - g \to 0$ in $L^\infty$ for some measurable $g$.

where we assume that $f'_n - g$ is in $L^\infty$ for all large enough $n$.

Is it true that $f$ is differentiable a.e. with $f' = g$ a.e.?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.

Let $f_n: [0, 1] \to \mathbb R$ be continuous functions that are differentiable almost everywhere. Suppose that there exist measurable functions $f, g$ such that

1. $f_n \to f$ uniformly,

2. $f'_n - g \in L^\infty$ for all $n \in \mathbb N$.

3. $f'_n - g \to 0$ in $L^\infty$.

Is it true that $f$ is differentiable almost everywhere, with $f' = g$ almost everywhere?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.

Let $f_n$ be a sequence of continuous, differentiable a.e. functions with $f_n \to f$ uniformly and $f'_n - g \to 0$ in $L^\infty$ for some measurable functions $f, g$, where we assume that $f'_n - g$ is in $L^\infty$ for all large enough $n$.

Is it true that $f$ is differentiable a.e. with $f' = g$ a.e.?

Note: We do not assume $f'_n, g$ to be in $L^\infty$ a priori, only their difference.