Let $\chi$ be the characteristic function of the rational numbers in $[0,1]$. Does there exist a sequence $\{f_n\}$ of continuous functions on $[0,1]$ that converges pointwise to $\chi$?
characteristic function of the rationals
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real-analysis
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3See also http://math.stackexchange.com/questions/15088/is-every-lebesgue-measurable-function-on-mathbbr-the-pointwise-limit-of-cont/15091#15091 and http://math.stackexchange.com/questions/4738/points-of-continuity-of-a-function-being-dense. – 2011-09-15
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2The characteristic function is a classic example of the Baire class 2 http://en.wikipedia.org/wiki/Baire_function. – 2011-09-15
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2Good questions here should have two additional pieces of information: where you encountered the problem, and what you have already tried. – 2011-09-15