sets where sequence of continuous functions diverge

Related Exercise from Rudin’s Real and Complex Analysis.

Exercise 5.20

• Does there exist a sequence of continuous positive functions $f_n$on ${ℝ}^1$ such that $\{f_n(x)\}$ is unbounded if and onlyif $x$ is rational?

• Replace “rational” by irrational in (a) and answer the resultingquestion.

• Replace “$\{f_n(x)\}$ is unbounded”by “$f_n(x)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$”and answer the resulting analogues of (a) and (b).

Solution:The answer to (a) is negative.This by showing thatthe subset of points where such sequence is unbounded must be $G_{\delta }$.But the rationals cannot be such,since in $ℝ$ dense $G_{\delta }$ sets must be of second category.

Rest of the answer not yet ready here