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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.

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be Riemann-integrable. The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.

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Show or contradict that $f$ is Rieman integrable.

Let $I=[a,b]$ be closed and bounded and $f:I\rightarrow\mathbb{R}$ be a monotone function show or contradict that $f$ is Riemann integrable. Any hint are welcome since I have no idea where to start.
R.vW
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Continuity of Riemann Integral Regarding Integrator

For $f \in R([a,b], \alpha)$ define $F(x) = \int_a^xfd\alpha, x\in [a,b]$ show that : i) $\alpha$ is continuous at some $x\in [a,b]\Rightarrow F$ is continuous at $x$ ii)$\exists \alpha'(x)$ and $f$ is continuous at $x$ for some $x \in…
Daschin
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Riemann integrability of $f(x) = 0$ if $x=0$ or $x = 1/n$ for some $n \in \mathbb{N}$, $f(x)=1$ otherwise

Let $$ f(x) = \begin{cases} 0 & \text{if $x=0$ or $x = 1/n$ for some $n \in \mathbb{N}$,} \\ 1 & \text{otherwise.} \end{cases} $$ Is this function Riemann-Stieltjes integrable in $[0,1]$? For the upper Riemann-Stieltjes integral, all…
snapper
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Riemann Integral, Monontone

Just a general question please, if a function is not monotone, can it still be Riemann Integrable? Constant functions are also Riemann Integrable, right? Thank you.
USERMATHS
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Riemann integrability of function

Let f a Riemann integrable function (in the proper sense) over [a,b]. Let g be another function that is different from f only in a finite number of points. How can I show that g is Riemann integrable and that the two integrals (of f and g) over…
Gmv93
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functions that are Riemann Integrable but its absolute value is not Riemann Integrable?

can anyone give me an example of functions that are Riemann Integrable on $[0, +\infty]$ but its absolute value is not Riemann Integrable on $[0, +\infty]$? Thanks!
Yaoxian Qu
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For any increasing functions $f : [a,b] \rightarrow R$, the set of discontinuities has measure zero.

Is this statement true or false? If true, how can I justify it with a proof. If false, is there a counter example. For any increasing functions $f : [a,b] \rightarrow R$, the set of discontinuities has measure zero.
SRC94
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Without measure, need $\int_a^b |f|^pd\alpha =0\Rightarrow \int_a^b |f|d\alpha=0$

As in the title, I want to show $\int_a^b |f|^pd\alpha =0\Rightarrow \int_a^b |f|d\alpha=0$ With measure theory, this implication is obvious. But using the definition, as a Riemann Integral, seems impossible to me right now. This material was so…
J. Moeller
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if f and g are riemann integrable on [a b] , g is non negative and f is bounded

I am trying to prove the following: if $f$ and $g$ are Riemann integrable on $[a,b]$, $g$ is non negative and $f$ is bounded then there exists $c$ such that $$\int_a^b f(x)g(x)dx=m\int_a^c g(x)dx+M\int_c^b g(x)dx$$ Thank you for your help.
Pol
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Continuous riemann integral

Let $f: [-1, 1] \to \mathbb R$ be a continuous function such $f(x) \ge 0$ for every $x \in [-1, 1]$ and $f(0)=1$ Show that the lower $\int_{-1}^1f>0$. This is about being continuous, am I correct? But that's how far how I've gotten.
repulsive23
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is f an rienmann intergrable

For a function to be Rienmann integral, its upper integral and lower integral need to be the same. I don't understand the function they have given here where $x=r/2^n$ thus don't know how to apply to values of the bounds to the functions. Also how…
K_uddin
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Estimating Riemann Inegrals

Let R be the rectangle given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and let f(x, y) = x + 5y. We want to estimate the integral ∬f(x, y) dx dy. a. Divide each side of the rectangle into three equal parts, and for each of the nine squares pick the sample point in…
KelKel23
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Riemann integrability for a sequence of functions

Task: Let $f_n: [0, 1] \rightarrow \Bbb R$, $n \in \Bbb N$, be a sequence of functions with $f_n(x) := 1$ for $n!x \in \Bbb Z$, $0$ otherwise. Show that $f_n$ is Riemann integrable. Question: We didn't do anything with Lebesgue, so I think this…
Marc Heger
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If $f$ is Riemann integrable on $[a,b]$ then is it integrable on every subset of $[a,b]$?

Main Problem. Let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. If $I\subseteq [a,b]$ then prove or disprove that $f$ is Riemann integrable on $I$. I know that if $I$ be a closed an bounded interval then we can say that $f$ is Riemann…
user170039
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Riemann-Integral

$f$ is a bounded and continuous function with $f:D \to (0,\infty)$ on the Jordan-measurable set. Prove that inequality $$\int_Df(x)dx\int_D{1\over f(x)}dx\ge(vol(D))^2$$ Can someone please help me with this? I really am stuck on this Problem. Any…
suugii19
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