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[parent] sets where sequence of continuous functions diverge (Derivation)

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.

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"sets where sequence of continuous functions diverge" is owned by yotam. [ full author list (2) ]
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Keywords:  sequence continuous functions converge unbounded rationals irrationals

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Cross-references: second category, dense, points, subset, negative, solution, irrational, rational, unbounded, functions, positive, continuous, sequence, complex analysis, real

This is version 5 of sets where sequence of continuous functions diverge, born on 2005-07-15, modified 2006-07-18.
Object id is 7226, canonical name is RelatedExerciseFromRudinsRCAnalysis.
Accessed 1479 times total.

Classification:
AMS MSC: 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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