sets where sequence of continuous functions diverge

Related Exercise from Rudin’s Real and Complex Analysis.

Exercise 5.20

  • (a)

    Does there exist a sequenceMathworldPlanetmath of continuousMathworldPlanetmath positive functions fn on 1 such that {fn(x)} is unbounded if and only if x is rational?

  • (b)

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

  • (c)

    Replace “{fn(x)} is unbounded” by “fn(x) as n” 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 Gδ. But the rationals cannot be such, since in dense Gδ sets must be of second category.

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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