Compositions of function and integrability (is this right?)

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Homework Help Overview

The discussion revolves around the integrability of compositions of functions, specifically examining the case where one function is piecewise continuous and the other is a characteristic function. The original poster seeks to understand whether their approach using a ruler function and a characteristic function is valid.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to use the characteristic function of the rationals as one function and a rational ruler function as another to explore integrability. Some participants question the continuity of the characteristic function and its implications for the original poster's reasoning.

Discussion Status

The discussion is ongoing, with participants exploring different definitions of continuity and integrability. Some guidance has been offered regarding the nature of the functions involved, but no consensus has been reached on a viable approach yet.

Contextual Notes

There is a focus on the definitions of piecewise continuity and the characteristics of the functions being discussed, particularly regarding their points of discontinuity. The original poster's initial assumptions are being re-evaluated in light of these definitions.

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Compositions of functions and integrability (is this right?)

Homework Statement

We know that if f is integrable and g is continuous then g\circf is integrable. Show to this is not necessarily true for piecewise continuity. We are given the hint to use a ruler function and characteristic function.



Homework Equations




The Attempt at a Solution

I decided to use the characteristic function of Q for g as it was the one example the book gave of a characteristic function that is not integrable and the rational ruler function for f.

The characteristic function of Q is defined as g(x)=1 when x is rational and 0 when it isn't.
The rational ruler function is defined as follows= If x=p/q is rational then f(x)=1/q and 0 when z isn't rational.

g(f(x))=1 when x is rational and 0 when it isn't so the composition is equal to the characteristic function for Q and we know that isn't integrable. I just want to know if this is right. It just seems too easy...
 
Last edited:
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but ins't your function g(x) nowhere continuous to start with?
 
lanedance said:
but ins't your function g(x) nowhere continuous to start with?

Yeah, I guess that's true. Darn it...back to the drawing board. Anyone have any suggestions as to what would work?
 
well the ruler function is only discontinuous at a infinite countable subset of points...

what is your definition of piecewise continuous?
 
lanedance said:
well the ruler function is only discontinuous at a infinite countable subset of points...

what is your definition of piecewise continuous?

f is continuous except at a finite number of points where it is discontinuous.
 
Anyone?
 
Does g(x)=the characteristic function defined on the interval (0,1] work?
 

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