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 “ $f_{n}(x)\to\infty$ as $n\to\infty$ ” 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	2013-03-22 15:23:34
Last modified on	2013-03-22 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