If f is Riemann integrable on [a,b] then prove that there is an equispaced partition Pn such that limU(Pn,f) = limL(Pn,f) = the value of the integral.

score 65 · Answer 11 · 2018-02-03 16:10:25

65

Views

I have some unclear things in mind related to the Riemann integrals theorem for the uniform convergent series and uniform convergent string.

Published on 03 Feb 2018 - 16:10

score 2.8k · Answer 12 · 2026-03-20 01:15:58

2.8k

Views

Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$

Published on 20 Mar 2026 - 1:15

score 124 · Answer 13 · 2018-02-03 16:39:29

124

Views

A confusion about a problem in Zorich 11.4.3 Problem 1.c

Published on 03 Feb 2018 - 16:39

score 771 · Answer 14 · 2026-03-17 08:30:27

771

Views

Two equivalent definitions Riemann integral

Published on 17 Mar 2026 - 8:30

score 17 · Answer 15 · 2026-03-17 00:58:17

17

Views

A confusion about the partial limits of $F(x) = \int_{I_{a,x}} f(t) dt$

Published on 17 Mar 2026 - 0:58

List Question

Conditional convergent improper Riemann integral vs. Lebesgue Integral

I have some unclear things in mind related to integrating a Dirichlet function.

How to show that the upper and lower Riemann integrals of a function, say $f(x) = -2x$, are equal?

Using Riemann criterion for integrability to prove integrability of the function $F$

Necessary and sufficient conditions for Riemann integral of $f: E\subset \mathbb{R}^n \to E$ exists, where $E$ is and subset of $\mathbb{R}^n$

Prove that $\lim_{n\to \infty} [\int_E f^n (x) dx)]^{1/n} = M$ where $M = \sup_{x\in E} f (x)$

Let $E \subseteq \mathbb{R}^n$, and $f \in R(E)$, then is the fact that $f$ is continuous on $E$ implies $\mu(E) > 0$?

Lebesgue-style-Lower-Riemann (or inner Riemann) integral. What is the space of integrable functions? Is it deficient?

Proof of $\big|\int_E (f\cdot g)(x) \text{d}x\big| \leq [\int_E |f(x)|^p\text{d}x]^{(1/p)} \cdot [\int_E |g(x)| ^q\text{d}x]^{(1/q)}$

If f is Riemann integrable on [a,b] then prove that there is an equispaced partition Pn such that limU(Pn,f) = limL(Pn,f) = the value of the integral.

I have some unclear things in mind related to the Riemann integrals theorem for the uniform convergent series and uniform convergent string.

Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$

A confusion about a problem in Zorich 11.4.3 Problem 1.c

Two equivalent definitions Riemann integral

A confusion about the partial limits of $F(x) = \int_{I_{a,x}} f(t) dt$

Trending Questions

Popular # Hahtags

Popular Questions