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Non existence of sequence of continuous functions

By John Khan in Uncategorized on February 22, 2017.

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.