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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24 views

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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25 views

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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42 views

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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50 views

Prove the following result: if $f(x) \geq g(x)$ for all $x\in I$, then $p.c.\int_{I}f \geq p.c.\int_{I}g$.

Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ and $g:I\to\textbf{R}$ be piecewise constant functions on $I$. Prove the following result: (a) If $f(x) \geq g(x)$ for all $x\in I$, then $p....
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25 views

uniformly continous function - upper limit by partition

The function $v:\Omega \subset \mathbb{R}^n_{x}\times \mathbb{R}_{t} \rightarrow \mathbb{R}^n$ is continous and has compact support. So it is even uniformly continous. From this property they follow ...
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1answer
22 views

When the piecewise constant integral independs of the partition's choice?

Proposition Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ be a function. Suppose that $\textbf{P}$ and $\textbf{P}'$ are partitions of $I$ such that $f$ is piecewise constant both with ...
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31 views

Ill-defined derivative?

This is a follow-up question to my previous question, Riemann integral interval confusion, from the same book. Below is a snippet of context from the book: Claim (I should probably be using $\...
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38 views

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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22 views

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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1answer
43 views

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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129 views

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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28 views

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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1answer
100 views

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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9 views

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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52 views

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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46 views

Prove that $\left|\Sigma^n_{i=1}c_i\Delta x_i - \frac{b^2}{2}\right| \leq \frac{1}{2}\Sigma^n_{i=1}(\Delta x_i)^2$ where…

Let $P$ be a partition of $[0, b]$ defined as $P = \{ 0 = x_0 < x_1 < > \ldots < x_n = b\}$, and let $c_i \in [x_{i-1}, x_i]$ for every $1 > \leq i \leq n$. Prove: $$\left|\Sigma^...
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1answer
49 views

Prove that for each $n \in \mathbb{N},$ the set $P = \{j/n\}$ is a partition of $[0,1]$

$$P=\left\{\frac{j}{n}:j=0,1,...n\right\}$$ I can't find anywhere in my book where it is proven that a given set is a partition. It just jumps to showing, using partitions, that a function is ...
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2answers
516 views

Quadratic Variation of a continuous function

I have got the following problem, but no idea where to start, so maybe one of you can help me out. We have got a continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)\neq f(1)$. The task is to ...
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36 views

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions? I know that by the definition of an Archmedean sequence of ...
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1answer
157 views

Proof that a function is Riemann integrable if for any $\epsilon > 0$ there exists a partition P such that: $U(P, f) − L(P, f) < \epsilon$

I am working my way through the proof of the following: Let $f$ be bounded on [a,b]. Then $f$ is Riemann integrable if and only if for every $\epsilon$ there is a partition on $[a,b]$ such that: $0 \...
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150 views

Why can’t improper integrals be defined directly using Riemann sums?

The standard way to define an improper integral of the form $\int_a^\infty f(t)dt$ is as follows. We first define the Riemann integral $\int_a^xf(t)dt$ for each $x>a$ in the standard way, i.e. ...
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1answer
214 views

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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36 views

$f$ is integrable iff the $U(f;P)-L(f;P)< \varepsilon$ whenever mesh(P)<$\delta$?

This is very close to what is shown in Rudin (Thm 6.6) but Rudin doesn't mention anything about a delta or a mesh in his theorem. We were supposed to show this in class yesterday but the professor ...
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2answers
122 views

How do we calculate integrals without knowing differentiation?

To calculate the integrals, we use the fundamental theorem of calculus. i.e for example to find $\int f(x)dx$, what we do is we find a function $g(x)$ such that $g'(x) = f(x)$. But, is there any way ...
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2answers
315 views

prove $L(f)\leq U(f)$

How exactly would I go about proving the following statement? Given $f:[a,b]\to\mathbb{R}$ show that $$L(f)\leq U(f)$$ where $$L(f)=\sup_{P\in\mathscr{P}}L(f,P) \text{ and } U(f)=\inf_{P\in\mathscr{P}...
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1answer
285 views

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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62 views

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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0answers
58 views

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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96 views

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 ...
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2answers
58 views

Lower and Sup Sums [Integration, partitions]

So i have this problem. Let $f(x)$= {$x^2$ if -3$\le$ $x$ $\le$ 1, $-2x$ if $1$$\lt$$x$$\le$$2$ And $P$={$x_0$,...,$x_n$} a partition of [-3,2]. If $1$$\in$[$x_i-1$,$x_i$], find $m_i$, $M_i$, i.e, $...
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2answers
146 views

Can you explain how to write the proof of Section 1.15 Exercise 4(a) of Apostol's Calculus

I am attempting to self-study Calculus from Apostol's book, but I have gotten stuck on the proof for Exercise 4(a). There are indeed other solutions, such as the below two links, but they don't ...
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1answer
172 views

Riemann Sums and Upper Sums for a discontinuous function.

This question is from Abbott's Understanding Analysis: "If $f$ is not continuous, it may not be possible to find tags for which $R(f,P)=U(f,P)$. Show, however, that given an arbitrary $\epsilon>0 ...
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1answer
165 views

Question about Tagged Partitions and $\delta$-fine partitions

Let $f:[a,b]\rightarrow \mathbb{R}$ where $|f|$ is bounded by $M>0$. Assume $f$ is Riemann Integrable. Let $\epsilon>0$ be arbitrary but fixed. Let $P_{\epsilon}$ be a partition of $[a,b]$ such ...
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3answers
595 views

Showing that g is integrable and $\int^b_a{f}$ = $\int^b_a{g}$

Let $f$ be integrable on $[a,b]$, and suppose g is a function on $[a,b]$ such that $g$($x$) = $f$($x$) except for finitely many $x$ in $[a,b]$. Show $g$ is integrable and $\int^b_a{f}$ = $\int^b_a{g}$....
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1answer
68 views

Embedding of weighted Holder space into $L^p$

Suppose we have a bounded metric space $(X,d)$ and a countable measurable partition $Q$ of $X$ (with respect to some probability measure $\mu$) such that $\mu(q)>0$ for all $q\in Q$. We know that ...
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0answers
53 views

For two partitions of an interval any subinterval of the one with the strictly smaller mesh is contained in one or two adjacent ones of the other.

If for an interval $[a,b]$ you have two partitions $Z_1: a = x_0 < \cdots < x_m = b$ and $Z_2: a = y_0 < \cdots < y_n = b$ where the mesh of $Z_1$ is strictly greater than the one of $Z_2$...
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2answers
109 views

Help with partition/width

Let $f:[0,2] \to \mathbb{R}$ be defined by $$f(x)= \begin{cases} 1,& \text{if } x=\frac{2}{n} \text{ for some $n \in \mathbb{N}$};\\ 0, & \text{otherwise}. \end{cases}$$...
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1answer
40 views

Bounded variation of $\frac1f$ when $\inf(|f|)>0$ & $f$ bounded variation

I want to show if $\frac{1}{f}\in BV[a,b]$ when $\inf(|f|)>0 \land f\in BV[a,b]$. I tried to find a partition that $V(\frac{1}{f},P)$ is upper-bounded using the partition that makes $V(f,P)$ ...
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1answer
169 views

Riemann Stieltjes integral. Prove that $\lim_{|P| \to 0} S(f,P,T) = \int_a^bfd\alpha$

Problem: Let $\alpha \in BV[a,b]$ and let $f$ be continuous on $[a,b]$. Prove that $\lim_{|P| \to 0} S(f,P,T) = \int_a^bfd\alpha$ Background: BV stands for bounded variation. I am using $|P|$ to ...
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2answers
42 views

Checking Riemann integrability for a function defined separately for rationals and irrationals in any given interval.

I was working on the following function, trying to find its Riemann integrability: $$f(x) = \begin{cases} x & x \in \mathbb{Q}\cap [-1,0] \\ -x & x \in \mathbb{R}\backslash \mathbb{Q}\cap [-1,...
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2answers
410 views

Non-standard partition for Riemann Sums?

I know I definitely saw an example of this in this site in the past, but I can no longer find it. In many (dare I say most?) Calc. I classes, if I, say, wanted to evaluate $$\int_{a}^{b}f(x)\text{ ...
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1answer
183 views

Difference between gauge and partition in Integration Theory

I'm studying generalized Riemann integral using Bartle's Introduction to Real Analysis textbook. What's the difference between gauge and partition? Seems like the same concept to me.
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0answers
73 views

How Many Points are in the Partition?

This is a theorem proven in my book. As usual, I covered up the proof of the theorem so that I might prove it myself. I came up with essentially the same proof, the only point of contrast being that I ...
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1answer
274 views

If the Partition $P\subset P'$, then $L(f, P)\leq L(f, P')$ and $U(f, P))\geq U(f, P')$

I have this proof but I don't really understand it, mathematically and visually (if that makes sense). So the definition is: A partition $P'$ is a refinement of the partition $P$ is $P\subset P'$. ...
2
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2answers
404 views

Piecewise constant integral is independent of partitions

I'm trying to prove the following proposition (from T.Tao's Analysis 1 book): Let $I$ be a bounded interval, $f\colon I\to\mathbb{R}$ function piecewise constant with respect to both P and P' ...
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1answer
208 views

Change of integration limits for uniformly continuous function

I have arrived at the inequality $$ V(\gamma) \le \sup\limits_{P\in\mathcal{P}[a,b]}\sum\limits_{i=1}^n\int\limits_{t_{i-1}}^{t_i} \left| \gamma'(s)\right|ds $$ where $\gamma(t):[a,b]\to\mathbb{C}$ ...
2
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1answer
859 views

Show $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^bfdx=\sum\limits_{x_{k-1}}^{x_k}\int\limits_{x_{k-1}}^{x_k}fdx$.

Let $f$ be integrable on $[a,b]$, and let Let $P = \{x_0,x_1,x_2,...,x_{n−1},x_n\}$ be any partition of $[a,b]$. Show that $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^...
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3answers
50 views

$f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},…, \frac{n-1}{n},1 \}$ of $[0,1]$.

Let $f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},..., \frac{n-1}{n},1 \}$ of $[0,1]$. Show that $S(P)=\frac{1}{3}+\frac{1}{2n}+\frac{1}...
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0answers
17 views

Is this right? Showing that if $S\subset \mathbb{R}^n$ arbitrary and $f$ a $C^r$ function then there is $A$ open such that $f$ is $C^r(A)$.

Let $f : S \to \mathbb{R}$ a $C^r$ function, where $S$ is any subset of $\mathbb{R}^n$. We say that $f$ is differentiable on $x_o \in S$ if there is $U_{x_0}$ and $g : U_{x_0} \to \mathbb{R}$ such ...
8
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0answers
313 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...