Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.

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Lesson Parent: 
The Logical Framework For Definite Integrals
Problem: 

Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.

Answer: 

It is true that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

To prove this we need to prove two things:

The proof is now complete.

Finally, from our original definition of the definite integral we know that

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So this proof can be written more succinctly as: The bounded function $f(x)$ on $[a,b]$ is integrable on $[a,b]$ if and only if <Sign in to see all the formulas>. This is oftentimes referred to as the Darboux Integrability Theorem.

This is a very important proof and we recommend seeing this lesson for a broader discussion of what this proof means for definite integrals.