Can Riemann integrable be defined using the epsilon delta non method?

[hongseok]

TL;DR Summary

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability using the idea of the epsilon-delta method.

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability using the idea of the epsilon-delta method.

 

[mcastillo356]

I wanted to define Riemann integrability using the idea of the epsilon delta argument.

Hi, @hongseok. We call $f$ Riemann integrable on $[a,b]$ if there exists $L\in{\mathbb R}$ so that for every $\epsilon>0$ there exists some $\delta>0$ such that

$$|S(f\,\dot{P})-L|<\epsilon,\qquad\forall{P}.\quad{||P||<\delta}$$

for any tag $\dot{P}$ on $P$.

Did I help you? Best wishes!

 

[FactChecker]

TL;DR Summary: Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability using the idea of the epsilon-delta method.

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability using the idea of the epsilon-delta method.

Intuitively, this seems good for finite integrals. For a formal proof, you would need to work through all the details using the precise definitions that you are given. I don't think that your approach will work for infinite integrals, but your official definitions may not allow those anyway, or there might be some way around those.

 

[PeroK]

Intuitively, this seems good for finite integrals. For a formal proof, you would need to work through all the details using the precise definitions that you are given. I don't think that your approach will work for infinite integrals, but your official definitions may not allow those anyway, or there might be some way around those.

For improper integrals to and/or from $\pm \infty$, the integral is defined as a limit of proper integrals (with finite bounds). This is not part of the basic definition of a Riemann integral itself.