Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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Discrete analogue of bounded variation

What kind of sequences $(a_n)\subset\mathbb{R}$ are expressible as the difference of two increasing sequences?
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15 views

Average of Convergent Sequences Proof [duplicate]

Show that if ($x_n$) is a convergent sequence, then the sequence given by the averages: $y_n$=($x_1$+$x_2$+...+$x_n$)/n also converges to the same limit. I know that for all $\epsilon$$>$0, ...
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34 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
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10 views

Compact set in the interior of a cone

Suppose compact set $S \subseteq R^n$ is in the interior of $x_0+C$, where $C$ denotes a solid convex cone in $R^n$ with the vertex at $0$. I am trying to prove that $\exists r>0$ such that $$S ...
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A variation on a problem of Polya and Szego

Among the various propositions on real series and sequences in "Problems and Theorems od Analysis I" Pt. I Chap. 4 by Polya and Szego, I noted n.178 at page 39 which implies what follows. Let ...
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2answers
26 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
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1answer
56 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
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3answers
44 views

Consider $F(x,y)=f(x+3y,2x-y)$…

If $f: \mathbb{R}^2\rightarrow\mathbb{R}$ where $F(x,y)=f(x+3y,2x-y)$ with $f$ is defferentiable and $\nabla f(0,0)=(4,-3)$ compute the derivate at the origin in the direction of unit vector ...
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1answer
19 views

Euclid's Lemma using FToA

I would really appreciate some help understanding the following passage from my Real Analysis text. I have a professor who uses inquiry based learning, which basically means we all stare at each other ...
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2answers
28 views

Find local extrema of the following function.

Find local extrema of the function $$u(x,y,z)=\sin x \cdot \sin y\cdot \sin z$$ with the condition $$x+y+z=\frac{\pi}{2};\; x,y,z>0$$ Can anyone give me pointers on how to solve this problem? ...
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Convergence rate of generalised Fourier series.

Consider a Sturm-Liouville system over an interval $[a,b]$: $$(p(x)y')' + (q(x) + w(x) \lambda) y = 0$$ Induced by this Sturm-Liouville system is a set of special functions that form a complete ...
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1answer
42 views

Problems with understanding analyticity

I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem). An analytic function, is a function that has a power series that ...
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46 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
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3answers
61 views

Does there exist any unbounded above function $f(x)$ such that $f(x)<\log(x)$ for all $x>M$

Does there exist any unbounded above function $f: \mathbb{R} \to \mathbb{R}$ such that there is some $M > 0$ such that $f(x)<\log(x)$ for all $x>M$? Mainly I observed the fact that $\log(x)$ ...
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24 views

having trouble with take $\displaystyle\lim_{\alpha \to -1}$

$\Gamma(0)=\infty$ I have : $$\displaystyle \lim_{\alpha \to ...
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23 views

In the geometrical interpretation for integration how lower and upper rectangular approximation are functions of natural number?

I've attempted to prove this in the following manner. Let Q be a subset of $P[a,b]$ which contains partitions of each order exactly once. Now, if we consider mappings $F:N \to Q$ defined by $F(n)=p$ ...
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1answer
35 views

$\lim_{x\to 0} \frac{f(x)}{x}=-1 \implies \lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$.

I'm trying to prove if $\lim_{x\to 0} \frac{f(x)}{x}=-1$, then $\lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$. I've tried everything, substitution, limit composition, etc. Anyone could help me to solve this ...
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1answer
26 views

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$.

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$. We know when $f$ and $g$ are both AC functions, the integration by parts is true. Is it ...
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18 views

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq ...
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60 views

Prove $b>1, r>0 \implies b^r > 1$

The proof is supposed to be extremely elementary (using on Rudin's Principles of Mathematical Analysis Chapter 1 material). This actually is not the main problem, but I have simplified that problem ...
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1answer
41 views

Help show all compact sets are closed in the compact complement topology

Given the usual topology $(\Bbb{R},\tau)$ on $\Bbb{R}$, define the compact complement topology as $\tau'=\{A\subseteq \Bbb{R}:A^C$ is compact in $\Bbb{R}\} \bigcup \{\emptyset \}$. The question is to ...
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1answer
34 views

Vector field left-invariant then also its respective flow?

I was wondering whether left invariance of a vector field $X$ to a respective Lie group $G$ (so $dL(a)(x)(X(x))= X(ax))$ is transfered to the respective flow defined by $\frac{d}{dt} \phi^{t}(x) = ...
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14 views

uniformly convergent sequence of differentiable functions, series of derivative of terms not convergent

I am attempting to come up with a uniformly convergent sequence of differentiable functions $g_{n}:(0,1)\to \mathbb R$ such that the sequence $\{g_{n}'\}$ does not converge. I was thinking that ...
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49 views

Is $\int_0 ^1 \frac{1-x^p}{1-x} $ ever rational for rational non-integer values of $p$?

It is well known that the $n$-th harmonic number $H_n$ has the integral representation $\int_0^1 \frac{1-x^n}{1-x}$. If we replace $n$ with rational non-integer $p$, do we ever get a rational outcome? ...
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2answers
44 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
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1answer
62 views

Real Analysis Folland 1.22a.)

Refer to exercise 18 here: 1.18 Let $(X,M,\mu)$ be a measure space, $\mu^{*}$ the outer measure induced by $\mu$ according to (1.12)(I will define this in the attempted proof), $M^{*}$ the ...
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1answer
17 views

Borel isomorphism and approximation of Borel space valued function

In Kallenberg's Foundations of modern probability, he defines a Borel space $(S,\mathcal{S})$ as a measurable space which is Borel isomorphic to a Borel subset $B\in\mathcal{B}([0,1])$, ie., there ...
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Fubini theorem counter example

While reading Kallenberg's proof of Fubini theorem in Foundations of modern probability, I realized that he first proved Tonelli's theorem, then apply Tonelli to $f_+$ and $f_-$, the positive and ...
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How to define a continuous map from $[0,1] $ onto/into $\mathbb R$? [on hold]

It is possible to define a continuous map from $[0,1]$ onto $\mathbb R$?
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1answer
38 views

What will I pay in month x if I pay 1/36 of balance each month? [on hold]

I have an open note at a bank that I pay 1/36th of the balance every month. I am looking for an equation that will allow me to know what my payment will be on x month.
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3answers
28 views

A question about the formulation of the definition of a limit for sequences

So I know the definition of a limit of a the sequence is: $a$ is a limit of a sequence $\{x_n\}$ if given $\epsilon>0$ there exists a positive integer $N$ such that $|x_n-a|<\epsilon$ for all ...
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2answers
163 views

How to check the set to be closed?

The set is obtained by removing the rational points from the interval $[4,7].$ How do I check to see if this set is closed in $\mathbb R$ ?
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Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x)

Here is the theorem in mathjax: A real value function $f$ is continuous at $x \in R$ iff whenever a sequence of real numbers $x_{n}$ converges to $x$ then the sequence $f(x_{n})$ $\rightarrow f(x)$. ...
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Ternary expansion and Cantor set

If $x$ has a ternary expansion $\sum \limits_{k=1}^{\infty}\dfrac{c_k}{3^k}$ where each $c_k\in \{0,2\}$ then $x$ belongs to Cantor set. Proof: Suppose $x$ has a ternary expansion $\sum ...
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37 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
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Is this a “locally surjective” function?

I quote the "locally surjective" part because I haven't found any reference of that concept, but it kind of fits what I mean. Let $f:\mathbb{R}^N \to \mathbb{R}^M, f \in C^1, x_0 \in \mathbb{R}^N : ...
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4answers
89 views

How I can evaluate $\lim_{(x,y) \rightarrow (0,0)} xy(\frac{1+xy}{x^3+y^3})^{1/3}$

I don't have idea how I can evaluate this double limit $$\lim_{(x,y) \rightarrow (0,0)} xy \left(\frac{1+xy}{x^3+y^3} \right) ^{1/3}$$ could you help me please! I try prove that $f$ is continuous: ...
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2answers
39 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
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33 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
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$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
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32 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
3
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2answers
47 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
2
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1answer
45 views

Sufficient conditions for this function being linear [duplicate]

Let $f$ be a real-valued function for which, for every real $x,y$: $$f(x+y) = f(x)+f(y)$$ Does this imply that $f$ is a linear function ($f(x)=a\cdot x$)? If $f$ is differentiable, I think the ...
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3answers
42 views

Function defined on a closed interval must be bounded?

Intuitively, if a function $f$ is defined on $[a, b]$, then it must be bounded. Is there a theorem for this? I remember reading something related to this, but not the details.
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Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
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continuity ,differentiablity& riemann integral [on hold]

1.g(x)={█(0,if x is irrational@x if xis ration ,)┤ find all points of at which f is continouos 2.let A& B be compact sets define A+B ={a+b│aϵA and bϵB} show that A+B is compact. 3.let f be ...
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1answer
13 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
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continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
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1answer
12 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
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26 views

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...