Question

Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$

score 751 · Answer 1 · 2020-07-02 16:58:27

751

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Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$

Published on 02 Jul 2020 - 16:58

score 177 · Answer 2 · 2020-07-03 17:45:58

177

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If $f\in C^1[0,1]$, then $\left|\int_0^1f(x)dx-\frac{1}{n}\sum_{k=0}^{n-1}f\left(\frac{k}{n}\right)\right|\leq\frac{\int_0^1|f'(x)|dx}{n}$

Published on 03 Jul 2020 - 17:45

score 147 · Answer 3 · 2026-03-25 20:53:32

147

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A function is integrable if and only if the nets of lower sum and upper sum converge to the same number.

Published on 25 Mar 2026 - 20:53

score 80 · Answer 4 · 2020-07-03 22:30:31

80

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Prove that $L(f,P)=\inf\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$ and $U(f,P)=\sup\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$

Published on 03 Jul 2020 - 22:30

score 495 · Answer 5 · 2026-02-23 04:49:58

495

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Common refinement with different intervals

Published on 23 Feb 2026 - 4:49

score 646 · Answer 6 · 2026-03-28 15:11:25

646

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Understanding the substitution theorem of Riemann integration.

Published on 28 Mar 2026 - 15:11

score 205 · Answer 7 · 2020-07-06 07:33:31

205

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Riemann integrable (University of hyderabad 2017 exam)

Published on 06 Jul 2020 - 7:33

score 283 · Answer 8 · 2020-07-07 00:31:49

283

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Rudin 6.6 and 6.7, Recover Riemann Integral Definition from Calculus

Published on 07 Jul 2020 - 0:31

score 132 · Answer 9 · 2020-07-10 08:19:33

132

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$(f(x)+c)^2$ is Riemann integrable, but $f$ is not Reimann integrable on $[0,1]$

Published on 10 Jul 2020 - 8:19

score 167 · Answer 10 · 2020-07-12 05:37:32

>Let $sgn:\mathbb{R}\to\mathbb{R}$ be the signum function, Let $f:[-1,1]\to\mathbb{R}$ be continuous. Show that $\int_{[-1,1]} f \delta sgn=2f(0)$

score 104 · Answer 11 · 2026-03-27 22:03:19

104

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Riemann integrability criteria

Published on 27 Mar 2026 - 22:03

score 85 · Answer 12 · 2020-07-12 17:38:08

Define $g$ by $g(q(n))=2^{-n}$ and $f(x)=\sum_{r\in \mathbb{Q}:r<x}g(r)$. Prove $F(x):=\int_0^x f(y)\operatorname{dy}$ is not differentiable at $q$

score 52 · Answer 13 · 2020-07-12 19:09:51

52

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Can some explain where the first inequality is derived from?

Published on 12 Jul 2020 - 19:09

score 113 · Answer 14 · 2020-07-13 01:15:18

Let $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Let $g:[-b,-a]\to \mathbb{R}$ be defined by $g(x):=f(-x)$. Show that $g$ is Riemann integrable

score 80 · Answer 15 · 2020-07-13 22:35:32

If $f,g\in\mathcal C^1[0,1],\,f$ monotone, and $g(x)>g(1)=g(0)$ on $(0,1)$, then $\int_0^1 f(x)g'(x)\,dx=0$ if and only if $f$ is constant

List Question

Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$

If $f\in C^1[0,1]$, then $\left|\int_0^1f(x)dx-\frac{1}{n}\sum_{k=0}^{n-1}f\left(\frac{k}{n}\right)\right|\leq\frac{\int_0^1|f'(x)|dx}{n}$

A function is integrable if and only if the nets of lower sum and upper sum converge to the same number.

Prove that $L(f,P)=\inf\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$ and $U(f,P)=\sup\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$

Common refinement with different intervals

Understanding the substitution theorem of Riemann integration.

Riemann integrable (University of hyderabad 2017 exam)

Rudin 6.6 and 6.7, Recover Riemann Integral Definition from Calculus

$(f(x)+c)^2$ is Riemann integrable, but $f$ is not Reimann integrable on $[0,1]$

>Let $sgn:\mathbb{R}\to\mathbb{R}$ be the signum function, Let $f:[-1,1]\to\mathbb{R}$ be continuous. Show that $\int_{[-1,1]} f \delta sgn=2f(0)$

Riemann integrability criteria

Define $g$ by $g(q(n))=2^{-n}$ and $f(x)=\sum_{r\in \mathbb{Q}:r<x}g(r)$. Prove $F(x):=\int_0^x f(y)\operatorname{dy}$ is not differentiable at $q$

Can some explain where the first inequality is derived from?

Let $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Let $g:[-b,-a]\to \mathbb{R}$ be defined by $g(x):=f(-x)$. Show that $g$ is Riemann integrable

If $f,g\in\mathcal C^1[0,1],\,f$ monotone, and $g(x)>g(1)=g(0)$ on $(0,1)$, then $\int_0^1 f(x)g'(x)\,dx=0$ if and only if $f$ is constant

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