sets where sequence of continuous functions diverge
Related Exercise from Rudin’s Real and Complex Analysis.
Exercise 5.20

(a)
Does there exist a sequence^{} of continuous^{} positive functions ${f}_{n}$ on ${\mathbb{R}}^{1}$ such that $\{{f}_{n}(x)\}$ is unbounded if and only if $x$ is rational?

(b)
Replace “rational” by irrational in (a) and answer the resulting question.

(c)
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 that the subset of points where such sequence is unbounded must be ${G}_{\delta}$. But the rationals cannot be such, since in $\mathbb{R}$ dense ${G}_{\delta}$ sets must be of second category.
Rest of the answer not yet ready here
Title  sets where sequence of continuous functions diverge 

Canonical name  SetsWhereSequenceOfContinuousFunctionsDiverge 
Date of creation  20130322 15:23:34 
Last modified on  20130322 15:23:34 
Owner  yotam (10129) 
Last modified by  yotam (10129) 
Numerical id  8 
Author  yotam (10129) 
Entry type  Derivation 
Classification  msc 26A15 
Classification  msc 40A30 