Unbounded and continuous almost everywhere

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Discussion Overview

The discussion revolves around the search for examples of functions that are continuous almost everywhere yet unbounded. Participants explore various functions and their properties, including integrability and specific definitions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests an example of a function that is continuous almost everywhere and unbounded.
  • Another participant proposes a specific function defined piecewise, which is continuous almost everywhere but unbounded.
  • A third participant expresses gratitude for the example provided.
  • One participant inquires about a function that is not Riemann integrable.
  • A different function involving a series and characteristic functions is suggested as a potential example of non-Riemann integrability.
  • Another participant suggests a simpler function, f(x)=1/x for x>0, claiming it meets the criteria of being continuous almost everywhere and unbounded.

Areas of Agreement / Disagreement

Participants present multiple examples and viewpoints, indicating that there is no consensus on a single example or approach. The discussion remains unresolved regarding the best example of such a function.

Contextual Notes

Some proposed functions may depend on specific definitions of continuity and integrability, and the discussion includes various interpretations of these concepts.

Ja4Coltrane
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Can anyone give me an example of a function f:[a,b]->R which is continuous almost everywhere yet unbounded?

Thanks!
 
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The function f:[0,1]->R given by
f(x) = n if x=1/n for some positive integer n
f(x) = 0 else
 
muchas gracias
 
Okay how about one which isn't Riemann (improper) integrable
 
[tex]\displaystyle\sum_{ k = 1 }^\infty k \chi_{ [ 0, \frac{ 1 }{ k^2 } ] }[/tex]
 
Um, surely f(x)=1/x for x>0 and f(0)=0 is far simpler, and does everything needed.
 

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