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Non existence of sequence of continuous functions

Prove that there does not exist a sequence of continuous functions f_n:[0, 1] \rightarrow \mathbb{R} such that converges pointwise, to the function \chi_{\mathbb{Q}} , where \chi_{\mathbb{Q}} is the characteristic polynomial of the rationals in [0, 1].

Solution

The indicator of the rationals is no other function than

\chi_{\mathbb{Q}}= \left\{\begin{matrix} 1& ,& x \in \mathbb{Q}\\ 0& , & \text{elsewhere} \end{matrix}\right.

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.

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