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# Non existence of sequence of continuous functions

Prove that there does not exist a sequence of continuous functions such that converges pointwise, to the function , where is the characteristic polynomial of the rationals in .

Solution

The indicator of the rationals is no other function than

It is known that pointwise limits of continuous functions have a meagre set of points of continuity. However, this function is discontinuous everywhere and thus we cannot expect a sequence of continuous functions to converge pointwise to it.