Calculus/The Riemann-Darboux Integral, Integrability criterion, and monotone/Lipschitz function

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The IntegralEdit

What we introduced last time:

 

 

Are actually known as the lower and upper integrals respectively. Both are simultaneously the integral provided that the function that they are built from satisfy the following condition, the definition of integrability (but not the integrability criterion).

A bounded function   on   is integrable if  .

And that's all the integral really is!

Yet ANOTHER return of that summing exerciseEdit

This is just a quick little question, probably trivial. What does this mean for   ?

IntegrabilityEdit

As previously defined we can prove the integrability of a function by noting that  

However, there is a much more useful way to prove that a function, or an entire class of functions, is integrable. This is the theorem called the Integrability Criterion:

A bounded function   is integrable on   iff for all   there exists a partition   of   such that  

You'll notice that it has properties similar to the definition of the limit.

Since this is a proof of necessity assume that   is integrable on   . Let   be given and say   . Also   . Now by the approximation property there exists a partition   such that   and a partition   such that   . If we take the refinement   and apply the properties of refinement that we proved before:

 

and

 

We can manipulate this to:

 

Now for the proof to finish up the biconditional...

Integrability criterion proofEdit

You do it. If for all   there exists a partition   of   such that   then   is integrable on   .

Some hints: show that it implies   for all   then show this is so only if   .