# Questions tagged [partitions-for-integration]

For questions on finding or constructing a partition of an interval to compute the Riemann integral or Riemann-Stieltjes integral.

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### Common refinement with different intervals

Let $[a,b]$ be an interval. A partition of $[a,b]$ is $N+1$ points such that: $$P:= \{x_1= a, x_2, \ldots,x_{N+1}= b \}.$$ Moreover, for an interval $[a,b]$ we can take two partitions $P^1, P^2$, and ...
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### Partition of Unity for Covering of the Unit circle

I am considering the unit circle $B_1(0)$ in $\mathbb{R}^2$ as a bounded lipschitz domain. Now I am looking for an explicit smooth partition of unity that is subordinate to some cover that corresponds ...
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### Prove that if the partition $\textbf{P}'$ is finer than $\textbf{P}$, then $f$ keeps piecewise constant

Let $I$ be a bounded interval, let $\textbf{P}$ be a partition of $I$, and let $f:I\to\textbf{R}$ be a function which is piecewise constant with respect to $\textbf{P}$. Let $\textbf{P}'$ be a ...
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### Approximating the Riemann integral for a certain product

Suppose that we have two continuous maps $f,g\colon I\to\mathbb{R}$ on the interval $I:=[0,1]$. Now suppose that $0=x_{0}<x_{2}<\ldots<x_{n}=1$ is some partition of $I$ and that for each $i$ ...
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### If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

I can figure out how to prove this, assuming C is positive, but I'm not sure how to take into consideration if C < 0. The sup of any partition of $Cf$ will simply be Csupf(x). This makes the proof ...
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### How should I partition this interval to utilize the left endpoint $\frac{\epsilon}{2}$?

In the problem above, since f is continuous and thus uniformly continuous on the compact interval $[\frac{\epsilon}{2},1]$, it is possible to directly refer to the general property that all uniformly ...
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### Limit of the ratio of two non-Riemann sums.

Let $\left[ {a,b} \right] \subset \mathbb{R}$ and $f,g:\left[ {a,b} \right] \to \mathbb{R}$ be two Riemann-integrable functions. Let $a = {x_0} < {x_1} < {x_2}... < {x_n} = b$ be a ...
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### Darboux Sum in terms of partitions

I've got a problem to solve, however during the lectures that was explained poorly. I was able to teach myself culculating double integrals. But that is more complex to understand. I will appreciate ...
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### Proving $\frac{1}{f}$ is Riemann Integrable

Suppose $f\in R(x)$ and $\frac{1}{f}$ is bounded on $[a,b]$. Prove that $\frac{1}{f}\in R(x)$ on $[a,b]$. We need to show $U(P,f)-L(P,f)\leq\epsilon$ to prove Riemann Integrability. To prove this ...
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### Probability of a conditional density function using the partition theorem

Given $f_x(x)=\frac{x}{2}$ (for $0 < x < 2$) and $f_{y|x}(y|x)=\frac{3y^2}{x^3}$ (for $0 < y < x$) determine $P(Y<3/2)$ I think it is $\int_0^2 P(Y<3/2|X=x)f_x(x)dx$ but I dont ...
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### Norm of a partition defined by a set, Riemann sum

The set $\{1,1.7,2,3.5,5\}$ determines a partition, $P$. Let $z_1=1$, $z_2=2$, $z_3=3$, $z_4=4$, and $f(x)=\sqrt{4-\left(x-3\right)^2}$. There’s three things I’m told to find: $\lVert P\rVert$ The ...
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### Prove $f$ is integrable on $[a,b]$ if $f$ has finitely many accumulation points on $[a,b]$.

Suppose interval $I\in[a,b]$ as finitely many limit points, $f:[a,b]\to \mathbb{R}$ is bounded on $[a,b]$ and continuous on $[a,b]\setminus I$. Use the fact that if $f$ is continuous except at ...
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### Help for this problem involving rieman integral and partitions

If $f: I--->\mathbb R$ is bounded, let $||f||:= Sup {|f(x)|: x \in I}$, and if $P =(x_0,...,x_n)$ is a partition of $I:=[a,b]$, let $||P||:=Sup [x_1-x_0,...,x_n-x_{n-1}]$ (a) If $P'$ is the ...
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### How to compete the proof of that for a map defined on an arbitrary subset $S$ of $\mathbb{R}^n$ , we can extend $f$ to $C^r$ map on $\mathbb{R}^n$

In the book of Analysis on Manifolds, by Munkres, at page 144, question 3, it is asked that Questions: First of all, I was writing this question to ask whether my proof was correct or not, but ...
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### What partial orders on tagged partitions generate the Riemann Integral?

Let $[a,b]$ be a closed interval in $\mathbb{R}$, and let $X$ be the set of tagged partitions of $[a,b]$. Now let’s define two partial orders over $X$. Let $P_1\geq_1 P_2$ if $P_1$ is a refinement ...