- #1

Adesh

- 735

- 191

- TL;DR Summary
- A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if for any ##\epsilon~~\gt 0## we can find a partition P of ##[a,b]## such that $$U(f,P) - L(f,P) \lt \epsilon$$

Integrals are defined with the help of upper and lower sums, and more number of points in a partition of a given interval (on which we are integrating) ensure a lower upper sum and a higher lower sum. Keeping in mind these two things, I find the following definition easy to

I'm having so many doubts in the above definition, what does it exactly mean to say "we can find a partition P such that ...", how can we ensure that we can find a partition P? And also this second definition doesn't say anything about what the integral will be. Please explain it to me how

*digest***A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if**

$$sup \{L(f,P) : \text{P is the partition of [a,b]}\} = inf\{U(f,P) : \text{P is a partition of [a,b]}\}$$

and integral of ##f## on ##[a,b]## is that same common number.However, it is said that the above definition is not handy and hard to use (I don't know why, I find it better) so we have another definition (derivable from the above one) which says$$sup \{L(f,P) : \text{P is the partition of [a,b]}\} = inf\{U(f,P) : \text{P is a partition of [a,b]}\}$$

and integral of ##f## on ##[a,b]## is that same common number.

**A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if for any ##\epsilon~~\gt 0## we can find a partition P of ##[a,b]## such that $$U(f,P) - L(f,P) \lt \epsilon$$**I'm having so many doubts in the above definition, what does it exactly mean to say "we can find a partition P such that ...", how can we ensure that we can find a partition P? And also this second definition doesn't say anything about what the integral will be. Please explain it to me how

**YOU PERSONALLY**understand that ##\epsilon## thing.