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Suppose we have a family of functions $\sum_{i=1}^n \alpha_i 1_{F_i}$ where $1_{F_i}$ is the characteristic function of $F_i \subset \mathbb{R}$, and $F_i$ is countable or $F_i^c$ is countable. This just gives us the constant functions with a countable number of discontinuities, but taking a finite number of values. Suppose we take the limit of such functions, say we have a monotonically increasing sequence of such functions $f_1 \leq f_2 \leq \ldots \to f$. What sort of function would $f$ be in general? My guess is $f$ would be equal to a constant function a.e., but with a countable number of discontinuities of arbitrary values, but I can't show this.

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$f$ can be any function which is constant almost everywhere and also has a greatest lower bound.

Let $f$ be such a function and let $a$ be its greatest lower bound. Then let $f_i(x) = f(x)$ if $-i \le x \le i$ and $f_i(x) = a$ otherwise: this sequence satisfies the chain of inequalities and converges to $f$. Conversely, no sequence exists which converges to a function which takes on arbitrarily large negative values, and no sequence exists which converges to a function with an uncountable number of discontinuities because the sequence itself is countable and $\aleph_0^2 = \aleph_0$.

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