It is known (by definition!) that the space of pointwise limits of continuous functions is the so called Baire class one functions, which can be characterized by their level sets (preimage of opens are all $F_\sigma$), or points of continuity (the Great Baire Theorem). It is not hard to see that any Baire class one function is a pointwise limit of a sequence of differentiable functions.

My question is: what are the functions $f:I\to \bf R$ which are pointwise limit of a sequence of differentiable $(f_n)$ with $(f'_n)$ also pointwise convergent to a function $g$ say (we don't ask any link between $f'(x)$ and $g(x)$ whenever the two make sense - this is already known: whatever can happen)? I wonder if the constraint on the derivatives adds somethings about $f$.

By using unbounded piecewise linear functions, on can get this way the characteristic functions of any segment, but already the case of the characteristic of open or closed sets is not clear (at least explicitly).

Take e.g. $F$ a closed subset of $[0,1]$: I thought of the use of an approximately continuous function $f_n$ defined as zero on $F$ and on $F_n = \{x; dist(x, F) \geq 1/n\}$, and by a constant otherwise, so that its integral on $[0,1]$ is $1$, but in the case of $F = [a,b]$ it gives a kind of cumulative characteristic function ($0$ on $[0,a[$, $1/2$ on $[a,b[$, $1$ on $[b,1]$).


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