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I am interested in the following question. Suppose $f: [0,1] \rightarrow [0,1]$ is a function (which may not be continuous).

Question When is $f$ the point-wise limit of a sequence of continuous functions? Is there a characterization of such functions within all functions?

For example, consider $f:[0,1] \rightarrow [0,1]$ which has $f(x) = 0$ if $x$ is rational and $f(x) = 1$ when $x$ is irrational. Can $f$ be point-wise approximated by a sequence of continuous functions?

user39598
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    Related: https://math.stackexchange.com/q/1001467/42969, https://math.stackexchange.com/q/549135/42969, https://math.stackexchange.com/q/1501274/42969 – Martin R Feb 16 '25 at 16:23
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    Look up Baire classes of functions (the pointwise limits of continuous functions are precisely Baire class $1$ ) - their discontinuities are limited (the function must be continuous in a lot of points from the Baire category point of view) your example is not there so cannot be approximated – Conrad Feb 16 '25 at 17:25

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