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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Upper Covering for a Compact Set

Show that if K is a compact set in $\mathbb{R}^n$, and U is an open set such that K $\subset$ U, then there exist $r>0$ and a finite collection of disjoint balls $\{B(x_j,r)\}_{j=1}^{N}$ such that ...
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4answers
22 views

An inequality, comprising of liminf and Sup of a set

The following was an example in a book called Applied analysis by Hunter. I am not exactly sure of what it really means or to how to approach it { $ x_{n,\alpha}\in \mathbb R $ |$ n \in \mathbb N $ ...
2
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1answer
136 views

Proof that square root of 2 exists and is a real number

In the proof for existence of $$\alpha^2 = 2$$ following steps are used: Choose $\alpha$ as upper bound for the set T $\{\alpha \in T : \alpha^2 < 2\}$ $$(\alpha + 1/n)^2 = \alpha^2 + ...
3
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1answer
46 views

Monotone convergence theorem in a special case

Suppose $C^{+}[0,1]$ be the set of all continuous functions with domain $[0,1]$ taking non-negative values only. Let $\lambda : C^{+}[0,1] \to [0,\infty)$ be a map that satisfies ...
3
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1answer
116 views

Example of tempered distribution

I am learning about tempered distributions. Today we learned that on $\mathbb{R}^n$, $\frac{1}{|x|^p}$ is a tempered distribution as long as $0 < p < n$. My question is, what goes wrong when $p ...
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1answer
49 views

Prove $\lim_{x \to c}{x^2}=c^2$ where $c$ is a real number [duplicate]

Prove $\lim_{x \to c}{x^2}=c^2$ where $c$ is a real number with the $(\epsilon, \delta)$ definition. I know that you need to assume a value for $\delta$. However, I don't understand how that one ...
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1answer
57 views

Is this a sufficient condition for differentiability

Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that, for all $c\in \mathbb{R}$, every vector in $f^{-1}(c)$ is supported by a unique hyperplane to $f^{-1}(c)$. Is $f$ ...
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4answers
405 views

Is there any way to prove this without logarithms?

I was given this problem: Show if $a>1$ and $n>1$ ($n$ and $a$ are integers) then, $\lim_{n\to\infty}a^{\frac{1}{n}}=1$ . The obvious solution is the following: Take the logarithm in base ...
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0answers
28 views

Continuous bijection of (0,1) onto a metric space that is not homeomorphic to (0,1). [duplicate]

Could someone suggest any please, I have no idea what function could it be.
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1answer
63 views

If $(f_n)_{n\geq 1} \to f$ uniformly and $(g_n)_{n\geq 1} \to g$ uniformly then $(f_n \circ g_n)_{n\geq 1} \to f\circ g$ uniformly

EDIT: Let $X$ be a compact metric space, $A$ a set. Let $(f_n)_{n\geq 1}$ be a sequence of continuous functions from $X$ to $\mathbb{R}$, and suppose $f_n \to f$ uniformly where $f:X\to \mathbb{R}$. ...
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2answers
101 views

Version of the Axiom of Induction for Real Induction?

Mathematical induction can be done using the axiom of induction, which is given as a formula written in the language of mathematical logic. Is there a way to express the ideas behind 'real induction' ...
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1answer
87 views

Fourier transform of a rational function with spike at origin

Consider a rational function $f(x) = \frac{p(x)}{q(x)}$, where both $p(x)$ and $q(x)$ are polynomial functions of the multivariate $x = (x_1, x_2,..., x_n) \in \mathbb{R}^n$. Also, let us say that the ...
10
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1answer
212 views

Linear differential equations of the $n$th order

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x;\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= \gamma_j\quad ...
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1answer
83 views

Linearity of the supremum

Short question: In which cases we have: Let $f_n$ be a sequence of functions, then $\sup_{k\in \mathbb N}(\sum_{n}^if_k(n))=\sum_{n}^i\sup_{k\in \mathbb N}f_k(n)$ ? I guess if the $f_k(n)$ are ...
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1answer
22 views

If the family of functions $H$ is relatively compact in $C[K]$, then it is also relatively compact in $L^p(K)$ for all $p \in [1,\infty]$

I am working through the book Fundamentals of Applied Functional Analysis, and in the chapter where the Ascoli-Arzela theorem is introduced, the question is asked: show that if the family of functions ...
3
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1answer
55 views

I need to show that the following function is positive.

I need to show that the following function is positive. $H(x)=2(7)^x+2(4)^x-2(3)^x+2(d+1)^x+2((d-2)(d+4))^x-2(d+2)^x-2((d-1)(d+5))^x$ Where $d=3,4,5 $ and $x\in[-1,0)$ From graph for different ...
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1answer
50 views

When using the method of characteristics, do we ever need the equation for the gradient?

I have a question about the method of characteristics (as defined in the PDE book by Evans). He summarizes the characteristic equations as (see eq. 11 on p. 98): (using $p=Du,z=u$ and the variable ...
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2answers
124 views

Transcendental numbers & logarithms [closed]

Given two coprime positive integers greater than one, say $n,\ m$ , where $n > m$ . How do we find the ratio $\dfrac{\log m}{\log n}$ in terms of $n$ and $m$ symbolically ? Claim: The ratio is ...
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1answer
24 views

Compact convex subset and hyperplanes

Suppose $K$ is a compact and convex subset and $x^*$ a point in $\mathbb{R}^n$. Suppose there exists $y\in \mathbb{R}^n$ such that $$\langle x^*, y\rangle> \sup_{x\in K} \langle x, y\rangle$$ ...
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1answer
78 views

Lagrange's Mean Value Theorem

This question has been posted before. I want to verify my solution. $f:[a,b]\rightarrow R$ be a continuously differentiable function s.t $f(a)=f(b)=0$. Prove that exists a point $c\in (a,b)$ such ...
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1answer
100 views

Why this function is Riemann integrable?

Let $f:[a,b]\rightarrow \mathbb R$ $$f(x)=0\text{ if x is irrational} $$ $$f(x)=\frac{1}{q}\text{ if }x=\frac{p}{q},\text{gcd}(p,q)=1 $$ When we consider the Dirichlet function, then we are saying ...
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1answer
104 views

Matrix with non-negative eigenvalues

Let $A \in \mathbb{R}^{n \times n}$ be a positive semi-definite $A \succcurlyeq 0$, and with positive diagonal elements ($A_{i,i} > 0$ for all $i$). Let $A$ have at least one eigenvalue equal to ...
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1answer
70 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
4
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2answers
142 views

Using second derivative to find a bound for the first derivative

Let $f$ be a twice differentiable function on $\left[0,1\right]$ satisfying $f\left(0\right)=f\left(1\right)=0$. Additionally $\left|f''\left(x\right)\right|\leq1$ in $\left(0,1\right)$. Prove that ...
0
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1answer
21 views

Uniform continuity of a $C^1$ function in $\mathbb{R}^n$ bounded away from a vector subspace of $\mathbb{R}^n$

Let $\mathbb{S}$ be a vector subspace of $\mathbb{R}^n$ and $d(x, \mathbb{S})$ be a distance function defined as $ d(x, \mathbb{S}) = \inf\{\|x-y\|: y \in \mathbb{S} \}, $ where $\|x\|$ is the ...
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3answers
745 views

Integral of an increasing function is convex?

Let $f$ be a real valued differentiable function defined for all $x \geq a$. Consider a function F defined by $F(x) = \int_a^x f(t) dt$. If f is increasing on any interval, then on that interval F is ...
2
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1answer
60 views

Can a monotone decreasing function intersect $y=-x$ on a set of more than measure $0$?

Can a monotone decreasing continuous function intersect $y=-x$ on a set of more than measure $0$ unless it equals to $y=-x$ on an interval? I am unsure whether something like Cantor's function can be ...
2
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1answer
66 views

Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
2
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2answers
142 views

Infinity norm of continuous function.

Let $f$ be a continuous function on the measure space $\mathbb{R}^n,\mathcal{L},\lambda$(Lebesgue measure). Prove that $\|f\|_\infty = \sup\{|f(x)|$ $|$ $x \in \mathbb{R}^n\}$ I saw same ...
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2answers
102 views

Convergence of a series with integral inside

I'm trying to determine whether the series $$\sum_{n=0}^{+\infty}\int_0^{\frac{\pi}{2}}\sin^n(x)dx$$ converges, diverges or is indeterminate. Since $\sin(x)\le 1\;\forall x \in \mathbb{R}$ and ...
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0answers
68 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$

Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier ...
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2answers
138 views

Is limit function of continuous even functions always continuous?

Limit function of Uniform Continuous functions (pointwise converge) is always continuous. It is the fact. Limit function of continuous functions is not always continuous (am I right?) However, for ...
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1answer
12 views

Local extrema of a function, considering its $n-th$ derivative where $n$ might be odd or even.

I found the following in my notes and need help to understand it. Consider the following theorem: "Let $A\subseteq \mathbb R^n$ be open, $f:A\to \mathbb R$. Suppose also that $f$ has all first order ...
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1answer
45 views

The dual space of a product space

Suppose $\prod^n L^1(I) $ is the product space of n $L^1$ integrable functions. What would be its dual space of continous linear functionals? would it be $\prod^n L^{\infty}(I)$? Do I need the norm ...
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1answer
35 views

Axiomatic proof help?

Can anyone help me to prove this using only the axioms? " If a is a positive real number which is less than the real number b, then the negative of the reciprocal of a is less than the negative of ...
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3answers
2k views

Why is zero the only infinitesimal real number?

I am currently reading Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler and was wondering if someone could help me with an aspect treated in the book. On page 24 he says a ...
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1answer
195 views

If $f$ and $g$ are monotone increasing and if the composite function $f\circ g$ is defined, then it is also monotone increasing.

For this question the definition for monotone increasing is that $f$ is monotone increasing if for all $x_1<x_2$ where $x_1,x_2\in I$, $f(x_1) \leqslant f(x_2)$. I have to apply that definition to ...
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2answers
47 views

question about Limit point of sequence

May be i am wrong! but i am confused in this. consider the sequence: $$<(-1)^n> = <-1, 1, -1, 1,...>$$ In the text, it is given that 1 and -1 are limit points of sequence $<(-1)^n>$. ...
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2answers
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If $a>1$, prove $\displaystyle{\lim_{x\to +\infty}a^x = +\infty}$ and $\displaystyle{\lim_{x\to -\infty} a^x = 0}$

so for this question some of the ideas I have is to just let $\epsilon > 0$ and for $x>N$ and now I have to find an N. My starting point is to somehow prove that $|f(x)-f(c)| < \epsilon$ so ...
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0answers
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Proof of Fermat's theorem on extremum of functions

Here's the statement of Fermat : "Let $f :\mathbb{R} \rightarrow \mathbb{R}$ (generally $f$ is defined on an open set), if $f$ is differentiable at $c \in \mathbb{R}$ and has a local extremum at this ...
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1answer
111 views

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: ...
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0answers
38 views

Mollifiers and Rates of Converegnce

I am interested in how quickly a.e. convergence happens to say: $|f(x) - f(x+h)|$. Originally, I thought I had proved something way too strong, but smoothed that out while typing this question up. ...
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0answers
35 views

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$?

From Folland Abstract Harmonic Analysis - Why is $\int f(x \xi) d\xi = \int f(x \xi^{-1}) \Delta_H(\xi^{-1}) d \xi$? This is on page 57. Here is the notation: $H$ is a closed subgroup of a locally ...
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3answers
150 views

Bolzano -Weierstrass Theorem and uniform Continuity

The following problem has hints, but I am unable presently to use it. Suppose $f$ is uniformly continuous on $(a,b]$, and let $\{x_n\}$ be any fixed sequence in $(a,b]$ converging to $a$. Show ...
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5answers
1k views

A finite set is closed

Question: Prove that a finite subset in a metric space is closed. My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next? ...
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1answer
45 views

Do simple functions play a role in differentiation?

Integrability of functions over measure spaces is often defined by means of simple functions. This makes me wonder whether or not the simple functions should play a role in studying derivative of ...
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1answer
106 views

Find all cluster points of the following sequence

This is a problem in Analysis I by Herbert Amann and Joachim Escher Problem For $n\in\mathbb{N}$, define $$a_n:=n+\frac{1}{k}-\frac{k^2+k-2}{2}$$ where $k\in\mathbb{N}\backslash\{0\}$ satisfies ...
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1answer
250 views

Cauchy's Mean Value Theorem. What can we say about $c$ with more information.

My questions is about Cauchy's Mean Value theorem which states: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there ...
4
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2answers
103 views

Estimate of a convolution from a paper by Michael Christ

I don't understand Lemma 2 of the paper Hilbert transforms along curves, II: A flat case by Michael Christ. The situation is as follows. I slightly simplified it from the exact context in the source. ...
7
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1answer
66 views

Making a set into a manifold

Let $n \in \mathbb{N}$, $M$ be a set and let $\mathcal{A} = \{(\varphi_a, U_a)\}_{a \in \mathcal{A}}$ be a system of tuples so that: $U_{a} \subseteq \mathbb{R}^n$ is open for all $a$; $\varphi_a: ...