# Questions tagged [riemann-integration]

The Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

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### prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx$

I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
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### Show that $f(x+1)$ is Riemann integrable [closed]

Let $f$ be a Riemann integrable function in $[a,b]$, show that $f(x+1)$ is Riemann integrable in $[a+1,b+1]$ I tried to use the Riemann criterion but I am not sure if this is the right way, because I ...
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### Integral of Cantor set indicator function [closed]

Say I have a function h(x) which is 1 whenever x is in the Cantor set and 0 for all other points in [0,1]. How can I prove this function is Riemann-integrable, and how can I show its integral is 0?
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### Why did the author write "Why?"? Any deep reason? ("Calculus Fourth Edition" by Michael Spivak)

THEOREM 6 If $f$ is integrable on $[a,b]$, then for any number $c$, the function $cf$ is integrable on $[a,b]$ and $$\int_a^b cf=c\cdot\int_a^b f.$$ PROOF The proof (which is much easier than that of ...
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### Use the definition to find $\int_0^2f(x)dx$ where $f (x )=\cases{ 1,&$x \ne 1$\\2,&$x = 1$}$

Use the definition to find $\int_0^2f(x)dx$ where $f (x )=\cases{ 1,&$x \ne 1$\\2,&$x = 1$}$ I am stuck with this question, we studying Riemann integration, in my mind I can do the integral ...
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### If all projections on the axis are integrable, do the iterated integrals exist?

Suppose $f(x,y) : [0,1] \times [0,1] \to \mathbb{R}_{\ge 0}$ is a function, not necessarily continuous, nor necessarily Riemann integrable on $[0,1] \times[0,1]$. By existence of an integral here we ...
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### Doubt in some steps in the Nirenberg-Gagliardo-Sobolev inequality.

I understood almost everything. But I got stuck in two steps that seems to be very simple. The first doubt is in the first inequality, in the beginning of the proof. In the first equality, for ...
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### Is this function equal almost everywhere to a Riemann integrable function?

Let $(a_n)_{n=1}^{\infty}$ be an enumeration of $\mathbb{Q} \cap [0,1]$. Define $B_n := \left\{x \in [0,1] : x \in \left( a_n \pm \frac{1}{2 \cdot 3^n}\right)\right\}$, $C := \bigcup_{n=1}^{\infty}B_n$...
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### Change of variable Theorem's proof for linear transformations

Let $U\subset\mathbb R^n$ be a rectifiable, bounded, open subset and $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ be a linear map. Let $f:\ T(U)\longrightarrow\mathbb R$. Suppose $T(U)$ is rectifiable,...
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### The Riemann Integral as the limit of a net

It is usually claimed that the Riemann integral can be seen as the limit of a net, but this is not clear to me. I can see that if the Riemann exists, the limit of the net must exist and they must ...
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### Prove Riemann integrability if $g(x) = f(x)$

I'm not sure if the following prove is correct. It seems incomplete. Hope you can help me: Theorem: Suppose $f:[a,b] \to \mathbb R$ is Riemann integrable. Suppose $g: [a,b] \to \mathbb R$ is a ...
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### Theorem on Riemann integral of monotone functions

I want to prove the following: Theorem: Let $(f_n)$ be a sequence of monotone (integrable) functions $f_n : [0,\infty) \rightarrow \mathbb{R}$ (for $t_i \leq t_j$ we have $f_n(t_i) \leq f_n(t_j)$ such ...
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### Why do Riemann sums converge?

I know from calculus that Riemann Integrals are Riemann sums. We use such integrals to calculate areas under curves. But why does adding an infinite amount of small terms give a finite result for non ...
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### why does $u$ lies in an interval $I_k=[x_{k-1},x_k]$ of $\dot P_1$ in Example 7.1.4, introduction to real analysis by Bartle & Sherbert?

I want to refer that similar question was asked here before (with no answers) Let $g:[0,3]\to \mathbb{R}$ be defined by $g(x)=2$ for $0\le x\le 1,$ and $g(x)=3$ for $1<x \le3.$ Let $\dot P$ be a ...
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### Riemann integrability of a charactersitic function

If $A$ is an open subset of $(0,1),$ then $\chi_A$ is Riemann integrable on $[0,1].$ My attempt: Since $A$ is an open subset of $(0,1),$ we can write $A$ as countable union of disjoint open intervals ...
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### Riemann integral of a function nonzero at countable points is zero

The problem is as follows. Let $E = \left\{\frac{1}{n}: n \in \mathbb{N}\right\}$ and let $f: [0,1] \mapsto \mathbb{R}$ be such that $f(x) = 1 \text{ for } x \in E$, zero otherwise. Show that $f$ is ...
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I am reading Spivak's book "Calculus". The definition of integrable given there is the following: $L(f,P)$ and $U(f,P)$ are the lower sum and the upper sum of $f$ for partition $P$. I ...