Let be a fixed integer consider the function
Prove directly from the definition of integrability that is integrable on
All I've got so far is that:
Any help? Thanks
One thing that should be obvious is that , since for all intervals in any partition you choose, will always equal . Proving that is a matter of choosing a clever partition. I don't think the "standard" partition (i.e. the one you're using) will work. You want a partition such that the total length of the intervals containing the can be made as small as desired.
Hey just came back to look at this problem.
So, when I use the above partition I get as the limit for both the upper and lower sum. This seems to violate my intuitive answer of
So my question is: can I just say, for some the sequence will get and stay at hence the limit of the upper sum is which implies the value of the integral is
Any faulty logic going on here?