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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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Is a function with limits Riemann integrable?

Suppose $f : [ 0, 1 ] → R$ and $\lim\limits_{x \rightarrow c} f ( x )$ exists for all $c \in [ a , b ]$. Show that $f$ is Riemann integrable on $[ a , b ]$. I can show this $f$ is bounded on $[a,b]$ ...
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Riemann integral is f integrable on I?

P={$0,(2-\frac{1}{n}), (2+\frac{1}{n}), 4$} and interval I=[0,4] Is f integrable on I? I have found that $U(f,P)$=$8+\frac{2}{n}$ and $L(f,P)$=$8-\frac{2}{n}$ So does this mean that we get bounds ...
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Integrating continuous function of two variables by one of the variables, do I get continuous function?

Let $f(x,y)$ be a function continuous in $x$ such that $g(x) = \int_a^b f(x,y) dy$ exists for every $x$. Is $g(x)$ necessarily continuous? I am especially interested in the Riemann and Lebesgue ...
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How to show Riemann integrability

Is the function $$f(x) = \begin{cases} \frac{x^2}{2}+4 &, x\ge0 \\ \> \frac{-x^2}{2}+2 &, x<0.\end{cases}$$ Riemann integrable in the interval$[-1,2]$? Does there ...
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Integrable function of which the antiderivative is not equal to the integral

Is there a Riemann-integrable function $f: [a,b]\to\mathbb{R}$ such that $f$ has an antiderivative which is not equal to $t \mapsto \int_a^t f(x)dx +c$ for any $c\in\mathbb{R}$?
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Is there $\int_ {0} ^ {1} g\,dx$?

Definition: A partition $ P $ of $ [a, b] $ is a finite set $ \{x_{0}, ..., x_ {n} \} \subseteq {[a, b]} $ such that $ a = x_ {0} <x_ {1} <... <x_ {n-1} <x_ {n} = b $. The norm of a $ P $ ...
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Show that there is a sequence $(P_n)$ of partitions of $[0,1]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g\colon[0,1]\rightarrow \mathbb{R}$, be defined as $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x)=1/x$ if $x \not\in\mathbb{Q}$. Show that there exists a sequence $(P_n)$ of partitions of $[0,1]$ ...
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For what type of functions do various types of Riemann sum approximations give exact answers? [closed]

For what kind of functions do each of the following approximation methods give exact answers: Mn, Ln, Rn, Tn, and Sn? Mn would be a midpoint Riemann sum, Ln would be a lefthand Riemann sum, etc.
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How to prove that Riemann integrable function is measurable?

If a function $f: A \rightarrow \mathbb{R}$ on a bounded set $A$ is Riemann integrable, how do you prove that $f$ is measurable? From partitions by $n$-dimensional intervals, I can make simple ...
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42 views

$ f\in\mathcal {R} _ {\alpha} ([a, b])$ and its equivalences

Definition: A partition $ P $ of $ [a, b] $ is a finite set $ \{x_{0}, ..., x_ {n} \} \subseteq {[a, b]} $ such that $ a = x_ {0} <x_ {1} <... <x_ {n-1} <x_ {n} = b $. The norm of a $ P $ ...
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If $f:A\subset \mathbb{R}^n\to\mathbb{R}$ bounded, $f(x)\geq 0$, for all $\epsilon>0$ exist P s.t. $S(f,P)<\epsilon$ then $f$ integrable, $\int_A f=0$

Problem: Given $f:A\subset \mathbb{R}^n\to\mathbb{R}$ a bounded function, $f(x)\geq 0$. Prove that if for all $\epsilon>0$ exist P partition such that $S(f,P)<\epsilon$ then $f$ is integrable ...
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Validity Check: Every function is integrable on its domain, regardless to continuity and differentiablitty.

F(x) is a function before I attempt to integrate such a function over a certain interval, what I do is check if the interval is a subset of the function's domain, if not then its not integrable. And ...
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Uniform convergence on compact subset and integration.

Let $(a,b)$ be a open interval and $f_n$ be a sequence of continuous functions on $(a,b)$. Given $f_n$ is uniformly convergent to $0$-function on any compact subset of $(a,b)$. I want to show that: $...
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Why is it necessary to split the definite integral of a piecewise function into a sum

The second fundamental theorem of calculus (Newton-Leibniz) tells us that: If $f$ is a real-valued function on a closed interval $[a, b]$ and $F$ is an antiderivative of $f$ in $[a,b]$ s.t. $F'(x)=f(...
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Arc length of a Polar curve as a Riemann sum

Suppose we have a curve in polar plane satisfying the equation $r=f(\theta)$ with $\theta\in[a,b].$ To find the area enclosed by this curve in this range of $\theta$ using Riemann integrals, we ...
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Reference request: Relation between Riemann and Lebesgue integral

I'm looking for a reference that gives the relations between Riemann and Lebesgue integrals. For example, it must contain facts like: (1) Let $f: [a,b] \to \mathbb{R}$ be a map. Then $f$ Riemann-...
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How to do this problem without using infinitesimal?

A rod of linear charge density a of length h, What will be the electric field at an axial point at a distance x from end of the rod (the end at which the origin is chosen for defining charge ...
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derivative exist almost everywhere

I am working on a problem finding an error of an example that uses integration by parts. The condition of the example says that $f'$ and $g'$ exist almost everywhere. If $f'$ and $g'$ exist almost ...
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Two-dimension Integral proof question

So I have two assumptions as follows: Let $S$ be a bounded subset of $\mathbb{R^2}$. For every $\epsilon>0$ there exists rectangles $R_1, ..., R_L$, s.t. the set $S \subset \cup_{i=1}^{L} R_i$ ...
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1answer
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Connection between u-substitution change of limit and Riemann sum

I worked with integration by $u$ substitution for a bit now, but I still have a hard time rationalizing why we change the limits of the integral when doing $u$ substitution (I know we don't have to ...
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1answer
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Understanding the Lebesgue criterion for Riemann-integrability

The Lebesgue criterion for Riemann-integrability of a function $f \colon \mathcal{D} \subseteq \mathbb{R} \longrightarrow \mathbb{R}$ states that a function is Riemann-integrable in a compact $\...
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What is the probability of a function being Riemann Integrable?

Suppose we are given a set $F =\{f|\ f:\mathbb{R} \to \mathbb{R}\}$, i.e. an arbitrary set (not sure about its countablility) of real-valued functions. What is the probability, that a function $f_k$, ...
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116 views

Prove that the interval [a, b] does not have measure zero

I want to prove that $[0,1]$ does not have measure zero but the book says $``$Explain why the following observation is not a solution to the problem: Every open interval that contains $[a,b]$ has ...
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Upper and lower Riemann sum of Partitions depending on $n$

I am currently self studying Riemann integrability and got to a part that states that if partitions P and Q are such that Q is a refinement of P then $$L (f,P) \leq U (f,Q) \leq U (f,Q) \leq U (f,P)$$ ...
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1answer
29 views

Riemann-integration problem

Here is the exercise Let $f:[a,b]\rightarrow \mathbb{R}$ be Riemann-integrable. Prove that $f^+$, $f^-$ and $|f|$ are also Riemann-integrable, when $$f^+=\begin{cases} f(x) & f(x)\...
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1answer
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Does $\implies wf(y)<\epsilon$ , mean $|f(y)-f(x)|<\epsilon?$

Right here, the poster claimed that for $\epsilon>0\;\text{and}\;x\in I,$ $$|y-x|<\delta\implies wf(y)<\epsilon$$ which implies that the $f$ is continuous at $x.$ Question: Does $\implies wf(...
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Let $f:[a,b]\to\Bbb{R}$ be a bounded function which is continuous except at a point $x_0\in [a,b]$. Then, $f$ is Riemann integrable over $[a,b]$.

Let $f:[a,b]\to\Bbb{R}$ be a bounded function which is continuous except at a point $x_0\in [a,b]$. Then, $f$ is Riemann integrable over $[a,b]$. My Proof Consider the restriction \begin{align} g:[...
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If $f$ is a derivative then is $|f|$ also a derivative?

If $f$ is a derivative then, is $|f|$ also a derivative? If $f$ is Riemann integrable then it's true. But, if it's not the case,then is it true?
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Prove that this function is Riemann Integrable on $[0,1]$ [duplicate]

Let $E:=\{\frac{1}{n}: n\in \mathbb{N}\}$. Define $f$ on $[0,1]$ by $f(x)=\begin{cases}1 ~~~~~~~~\text{if $x\in E$}\\ 0~~~~~~~~\text{if $x\notin E$ }\end{cases}$. Show that f is Riemann ...
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Riemann-integral for a function that has infinitely many discontinuity points

The question is following: Let $f:[0,1]\rightarrow \mathbb{R}.$ $f(x)=x,$ if $x=1/n, n\in\mathbb{N}$ $f(x)=0,$ otherwise. Is $f$ Riemann-integrable? If it is, what is its value? I know that the ...
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1answer
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Show that $f(x)=0,\;0\leq x<1/2,\; f(x)=1,\;1/2\leq x\leq 1$ is Riemann integrable over $[0,1]$ and find its value.

I want to prove that \begin{align} f:[0&,1]\to \Bbb{R}\\&x\mapsto \begin{cases}0,&0\leq x<1/2,\\\\ 1,&1/2\leq x\leq 1. \end{cases}\end{align} is Riemann integrable over $[0,1]$ ...
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1answer
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On anti-derivative of functions [closed]

Let $f,g,h: \mathbb R \to \mathbb R$ be differentiable functions. (1) Does there necessarily exist a differentiable function $F: \mathbb R \to \mathbb R $ such that $F'=\max \{f' ,g' \}$ ? (2) Does ...
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Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique

The density function of the distribution function of a continuous random variable is not uniquely defined. A new density function can be obtained by changing the value of the function at finite ...
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Riemann Integral of discontinuous function

So I know you can integrate some discontinuous functions when the function is discontinuous at a finite number of points. So you can integrate for example this function on a interval [-1,3]: $$ f(x):=...
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1answer
29 views

Logical mistake in a proof of Dirichlet's test

While writing a proof of Dirichlet's Test, [ in which, a required condition is that $\displaystyle{\int_a^B\phi(x)dx}$ is bounded $\forall B>a]$, I made a logical mistake, which I noticed upon ...
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Clarification required for the proof of Riemann Lebesgue Lemma

Riemann Lebesgue Lemma Proof. In the proof given above, I came across an unfamiliar notation $g= \sum m_i \chi_{[x_i-1,x_i]}$. What does the $\chi$ mean here? Is it the characteristic function of the ...
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1answer
44 views

Prove that if $f$ is continuous at $x_0\in [a,b]$ and $f(x_0)\neq 0,$ then $\sup\limits_{P}L(|f|,P)>0$

Can you help me check if this proof is correct? If not, kindly provide a better proof Prove that if $f$ is continuous at $x_0\in [a,b]$ and $f(x_0)\neq 0,$ then $\sup\limits_{P}L(|f|,P)>0$ ...
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1answer
48 views

A case where Lebesgue integrable implies Riemann integrable

Let $I$ an interval on $\mathbb{R}$ such as $I=(a,b)$, with $a$ or $b$ could be equal to infinity. And we have $f\in \mathcal{L}^1(I,\mathcal{B}(I), \lambda)$, then do we have always $$\int_{(a,b)}...
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1answer
31 views

Are there any numerical integration methods that do not involve rectangles or polynomial approximations?

For the Riemann integral, are there any methods of numerical integration that do not involve rectangles or approximating the area with a polynomial function? I am aware of the trapezoidal rule, but I ...
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306 views

When is a Lebesgue integrable function a Riemann integrable function?

When is a Lebesgue integrable function a Riemann integrable function ? And if we have $f\in \mathcal{L}^1([0,1],\lambda)$, does it implies that $f$ is Riemann integrable, and why ?
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Let $f$ an a.e continious and bounded function, is $f$ Riemann integrable function? [closed]

If we have an almost evrywhere continious and bounded function $f$ on $[0,1]$, can we deduce that $f$ is Reimann integrable ?
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Right Riemann sum Error bound proof

How do we prove the right Riemann sum error bound? In wikipedia (https://en.wikipedia.org/wiki/Riemann_sum#Right_Riemann_sum) they have mentioned the following bound, but no proof. $$\left | \int_a^bf(...
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Proof of second Fundamental theorem of calculus

Is there any other proof of this? Second fundamental Theorem of Calculus: If $f$ is differentiable on $[a,b]$ and $f'$ is integrable on $[a,b]$, then $$\int^{b}_a f'(t)dt=f(b)-f(a)$$ My proof ...
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1answer
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Prove that a Jordan measurable set of measure 0 is Riemann integrable

My problem is: I have a set $A$ which is Jordan measurable, and I have a function $f:A\rightarrow\mathbb{R}$. I need to prove that if the measure of A is $0$, then $f$ is Riemann integrable and ...
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2answers
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Proof of first Fundamental theorem of calculus

Can you please, check if my proof is correct? Suppose that $f:[a,b]\to \Bbb{R}$ is continuous and $F(x)=\int^{x}_{a}f(t)dt$, then $F\in C^{1}[a,b]$ and $$\dfrac{d}{dx}\int^{x}_{a}f(t)dt:=F'(x)=...
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1answer
54 views

Disprove: If $f$ is Riemann integrable on $ [a, b]$, then $f$ is continuous on $[a, b]$.

Do you have any other function that serves as a counterexample to this statement? If $f$ is Riemann integrable on $ [a, b]$, then $f$ is continuous on $[a, b]$. My counter-examples Consider the ...
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2answers
193 views

Example where The Lebesgue Integral is Better

What is an example that involves a fuction on an interval of the real numbers where the Lebesgue integral is better than the Riemann integral. By better, it probably means that the Lebesgue ...
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4answers
94 views

Find $\lim\limits_{n\rightarrow\infty}\int\limits_0^1f(x^n)dx$

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Show that for each $\epsilon\in(0,1)$, $\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon)f(0)$ ...
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2answers
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Riemann Integral: prove $\{U(P,f):P$ is a partition of $[a,b]\}$ is bounded below

[Defining the Riemann Integral: ] We consider a partition, $P=\{a=x_0<x_1<...<x_n=b\}$ of $[a,b]$ and a bounded function $f:[a,b]\rightarrow\mathbb{R}$. Next, we define- $$M_i=\sup\{f(x)|x\in[...
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1answer
47 views

$\underset{n\rightarrow \infty}{\lim}\int_{0}^{b_{n}}f_n(x)dx=\int_0^1f(x)dx$

Let $(f_n)_{n\in\mathbb{N}}$ ne a sequence of continuous functions on $[0,1]\rightarrow\mathbb{R}$ that converges uniformly to $f:[0,1]\rightarrow\mathbb{R}$. Let $(b_n)_{n\in\mathbb{R}}$ be an ...