Compositions of function and integrability (is this right?)

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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...

 

[lanedance]

but ins't your function g(x) nowhere continuous to start with?

 

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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?

 

[lanedance]

well the ruler function is only discontinuous at a infinite countable subset of points...

what is your definition of piecewise continuous?

 

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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.

 

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Anyone?

 

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Does g(x)=the characteristic function defined on the interval (0,1] work?