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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22 views

Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with $d(x_n, y_n) \to 0$.

Suppose $(X, d), (Y,\bar d)$ are metric spaces, $f:X \longrightarrow Y$. Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with ...
2
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1answer
35 views

Which exponents r>0 is the limit finite

I am trying to find values of $r>0$ such that $\lim\limits_{n\rightarrow \infty} \sum\limits_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r}$ is finite. I have tried to use integral methods for this limit such ...
0
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0answers
22 views

summation of $\sum_{k=0}^{\infty}{q^{\sum_{i=0}^{k}{n^{i}}}}$

Let $q\in (0,1)$ be fixed. Consider the sequence $\{q^{\sum_{i=0}^{k}{n^{i}}}\}_{k=0}^{\infty}$, where $n$ is a fixed odd positive integer. This sequence is convergent to zero by dini's theorem. set ...
0
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2answers
30 views

How can we find the bounds of the following integrals?

How can we find the bounds of the following integrals? $$\int_{0}^{+\infty}\int_{0}^{+\infty}\phi(x,y)dxdy$$ where $\phi(x,y)=$ $\begin{cases} 1 \quad if \quad x=y>0 \\ 0 \quad otherwise ...
1
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1answer
28 views

Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
2
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0answers
26 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
4
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3answers
111 views

In Tao's proof of the Hölder’s inequality

(Hölder’s inequality) Let $f \in L^p$ and $g \in L^q$ for some $0 < p,q \leq \infty$. Then $fg \in L^r$ and $\|fg\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^q}$, where the exponent $r$ is defined by the ...
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0answers
11 views

$\{(x,y) \subseteq \mathbb{R}^m \times \mathbb{R} | x \in A, y \in [0,f(x)]\}$ is J-measurable $\iff$ $ f: A \rightarrow \mathbb{R}$ is integrable

$\{(x,y) \subseteq \mathbb{R}^m \times \mathbb{R} | x \in A, y \in [0,f(x)]\}$ is Jordan-measurable $\iff$ $f:A \rightarrow \mathbb{R}$ is integrable, $f$ limited, $f \geq 0$ . I'm gonna call ...
0
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1answer
15 views

Show that f(x,y)=min(|x|,|y|) is not diferentiable at 0. [on hold]

I proved that the partial derivatives at 0 exists and equal 0. I do not know how to proceed.
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2answers
47 views

find a nonconstant function $f$ such that $|f(x) - f(y)| \leq |y - x|$ and part b)

a) find a function $f$, other than a constant function such that $|f(x) - f(y)| \leq |y - x|$ b) Suppose $f(y) - f(x) \leq (y - x)^2$ Prove that $f$ is a constant function. Solutions: I had troubles ...
0
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0answers
16 views

Version of Weierstrass nowhere differentiabele function

I am trying to proof that the function $$ f(x)= \sum_{n=1}^\infty \frac{2^n\sin (2^n x)}{3^n} = \sum_{n=1}^\infty f_n(x)$$ is nowhere differentiable. For fixed $x$ I have tried to construct a sequence ...
0
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1answer
28 views

Double sequences and iterated limits

Let a double sequence be defined as a function from $\mathbb{N} \times \mathbb{N} \to \mathbb{R}$, which we write as $(a_{m, n})$. We say that $(a_{m, n})$ converges to $a$, in symbols $(a_{m,n}) \to ...
0
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1answer
38 views

Show that there is a bounded linear functional $\ell : \mathscr C [0,1]\to\mathbb R$ with $\lVert \ell \rVert \leq 1,\ \ell(1)=0,\ \ell(\cos(x))=1$.

The title says it all. I've been assuming that the best way to do this is constructively, by finding such an $\ell$. I have by a theorem in our class that since $\lVert \cos(x)\rVert =1$, we know that ...
0
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2answers
37 views

Sequence of Radon Measures $\mu_n$ on $\mathbb{R}$

Problem: Find a sequence of signed Radon Measures $\mu_n$ on $\mathbb R$ such that $\langle \mu_n, \phi \rangle \to 0$ for every $\phi \in C^1_c(\mathbb R)$, and $|\mu_n|([0,1]) \to +\infty$. ...
0
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1answer
27 views

Three questions on measurable functions and $L^p$ spaces

I'm learning about measure theory and $L^P$ spaces and need help with the following questions: True or False (justify): $(1)$ Let $f:(-1, 1) \to \mathbb{R}$ measurable on $(-n, n), \; \forall ...
4
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0answers
45 views

minimize the area of convex hull of sum of 3 balls

How should we place 3 balls $B_1,B_2,B_3$ on the plane, if we want to minimize the area of convex hull of $B_1\cup B_2\cup B_3$ ? Balls can have boundary common points only -- the intersection of any ...
0
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3answers
52 views

Taylor series for $\ln(a-x)$ centered at $x=0$

Hey can someone help with finding the Taylor series for $\ln(a-x)$ I have tried using induction to determine the $f^{n}(x)$ $$f'(x)=-\frac{1}{(a-x)}$$ $$f''(x)=-\frac{1}{(a-x)^2}$$ ...
1
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1answer
50 views

Euclidean distance between $x\in\mathbb{R}$ and $\{x\in\mathbb{R} \mid f(x)=0\}$ [on hold]

Is there a generic formula to calculate the distance between an arbitrary real number $x\in\mathbb{R}$ and $$\{x\in {\mathbb{R}}\mid f(x)=0\}$$ where we have little information about $f$? In fact, my ...
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0answers
10 views

Every measurable subset of measure space is new measure space

Let $(X,\mathfrak{M},\mu)$ be a measure space and let $E\in \mathfrak{M}$. Prove that $E$ is also measure space. Proof: $(E,\mathfrak{M}_E,\bar\mu)$ be a measure space where $\bar \mu$ is "old" ...
0
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1answer
28 views

$\mathcal{L}^N(B_r(x)\cap E)> 0 \hspace{0.6cm} \forall r>0$ if every point is a Lebesgue Point

Exercise: Let $E$ be a Borel set such that every point is a Lebesgue Point for $\chi_E$ , and let $x \in \partial E$ (the topological boundary). Show that $\mathcal{L}^N(B_r(x)\cap E)> 0$, and ...
0
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0answers
6 views

Properties of $\nabla T_k(x)\cdot \nabla T_i(x)$ for a diffeomorphism $T$

Let $T:A \subset \mathbb{R}^n \to B \subset \mathbb{R}^n$ be a smooth diffeomorphism between $A$ and $B$. Is there anything I can say about the quantity $$\nabla T_k(x)\cdot \nabla T_i(x)$$ where ...
0
votes
2answers
32 views

Prove that $\iint_{\Bbb{R}^2}f(x^3+x,{y\over 3x^2+1})=\iint_{\Bbb{R}^2}f(x,y)$,

Prove that $\displaystyle \iint_{\Bbb{R}^2} f\left(x^3+x,{y\over 3x^2+1}\right) \, d(x,y) = \iint_{\Bbb{R}^2} f(x,y) \, d(x,y)$ for every continuous $f:\Bbb{R}^2\to \Bbb{R}$ with bounded support. My ...
2
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0answers
22 views

Constraints on a Chebyshev series representation of a CDF

My question is about deriving constraints for coefficients of a Chebyshev series which represents a CDF. Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
0
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1answer
30 views

Show that if $|s_{mn}-S|<\varepsilon$ then $|\lim_{n\to\infty}s_{mn}-S|\le\varepsilon$

This is the exercise 2.8.5 of the book Understanding analysis of Abbott. To put you in context I have that The sequence of partial sums $(s_{mn})$ is absolutely convergent, and the definition is ...
3
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3answers
81 views

$f \in C(\mathbb R)$ such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ ; is $f$ increasing?

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ , then is it true that $f$ is ...
0
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2answers
36 views

Using Mean Value Theorem to solve an inequality involving $\cos^{-1}(x)$

Question: Using the Mean Value Theorem, show that for all $-\frac{1}{2}\lt a,b \lt \frac{1}{2}$ with $a\lt b$ $$\lvert \cos^{-1}(a)-\cos^{-1}(b)\rvert \lt \frac{2\sqrt{3}}{3}\lvert ...
1
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1answer
46 views

Proof of L'Hopital's rule

I'm attempting to prove L'Hopital's rule. My solution so far is the following: Let $f,g:(a,b)\to\mathbb{R}$ be differentiable and continuous on $[a, b]$ with $f(a)=g(a)=0$ and $g(x)\neq0$ for ...
0
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2answers
19 views

Show pointwise convergence

I'd like to show that $\sum_{n=0}^{\infty}e^{-(x-n)^2}$ converges pointwise I can see that for $x=0$ the sum can be written as a geometric sum which then convergences, but I don't know how to ...
0
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1answer
41 views

Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
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0answers
24 views

integration of a non-negative continous functions

Let $f:[0,1] -> \mathbb R$ be continuous such that $f(t) \geq 0$ for all $t$ in $[0,1]$. Define $g(x) = \int_{0}^{x}f(t)dt$ then a) $g$ is monotone and bounded. b) $g$ is monotone, but not ...
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0answers
25 views

infinite series convergence test 2 [on hold]

Test the convergence of the series $\sum_{0}^{\infty}e^{-n^2}$ Does the integral $\int_{-1}^{1}\sqrt{\frac{1+x}{1-x}}dx$ exist?
1
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1answer
14 views

Infimum inequality comparing restrictions.

Suppose $f$ is a continuous function on the real line. Say we have two collections of sets $\{A_k\}_{k=1}^{n}$ and $\{B_k\}_{k=1}^{m}$, where $n>m$ and \begin{align} \bigcap_{k=1}^{n} A_k &= ...
0
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2answers
44 views

Question on Rolle's theorem involving roots

Use Rolle's theorem to show that $f(x)=x^3-\frac{3}{2}x^2+\lambda$, $\lambda \in \mathbf{R}$ never has 2 zeroes in $[0,1]$. I started by assuming that $\exists$ $2$ zeroes in$[0,1]$ Then ...
1
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0answers
21 views

Compact resolvent proof

I want to prove the following proposition: We consider the operator $$A=A_V=-\Delta+\frac{1}{4}|\nabla V|^2-\frac{1}{2}\Delta V$$(where $V$ is a polynomial) with domain ...
0
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1answer
26 views

Product of Lebesgue-null-set and arbitrary Lesbesgue-set is a Lebesgue-null-set again

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
2
votes
3answers
99 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
1
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1answer
24 views

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I'll first define what I mean by a 'component of a topological space': For a topological space $X$, write ...
0
votes
0answers
23 views

Harmonic function using Green's identity

Let$\ u:\mathbb{R}^{3}\rightarrow \mathbb{R}\ $ be a harmonic function and $\ 0< \varepsilon < \frac{1}{2}\ $ such that the following holds: $$\left | z \right |^{1-\varepsilon }\left \| ...
0
votes
1answer
20 views

Is the integral test for convergence still applicable?

$$\sum _{n=0}^{\infty \:}\left(n\ e^{-n^2}\right)$$ Can I still use the integral test to determine whether this series converges or diverges given that $f(x) = x\ e^{-x^2}$ is not decreasing on the ...
1
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3answers
82 views

Prove that $\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$

Let $(X,\mathfrak{M},\mu)$ be measure space. Let $f\geq 0$ be measurable function. Prove the following equality: $$\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$$ I can show only that $\int ...
0
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0answers
18 views

Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
0
votes
1answer
32 views

Lebesgue integral, path connected and compact

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
1
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1answer
46 views

Limit of a Riemann sum: $\lim_{n\to\infty} {n^5 \sum^n_{r=0}\frac1{(n^2+r^2)^3}} $

Required to find $\lim_{n\to{\infty}} {n^5 \sum^n_{r=0}\frac{1}{(n^2+r^2)^3}} $ $\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{n^2}{n^2+r^2})^3$ $\lim_{n\to{\infty}} \frac{1}{n} ...
1
vote
1answer
43 views

Discuss convergence of integral

Discuss the convergence of $\int_0^1 e^{\frac{1}{x}} \, dx$ Initially, I started with $$e^x \ge 1+x$$ Taking $\ln$ on both sides: $$x \ge \ln(1+x)$$ $$\frac{1}{x} \le \frac{1}{\ln(1+x)}$$ ...
1
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0answers
15 views

Prove $g(y) = \int_{\mathbb{R}} \sin(y^2x)f(x) dx$ is bounded and continuous on $\mathbb{R}$ for $f \in L^1(\mathbb{R})$

This question is from a practice qualifying exam. Here's my attempt (I'm a bit stuck on the continuity part): Since $f \in L^1(\mathbb{R})$, $f$ is bounded. Then: $$|g(y)| = |\int_{\mathbb{R}} ...
0
votes
1answer
43 views

Differential equation Cauchy problem resolution.

I can't find my mistake in solving this problem \begin{cases} y'(t) = y(t)/t + 2t(y(t))^2 \\ y(1) = 4 \end{cases} I recognize this as a Bernoulli equation and thus apply the substitution $z(t) = ...
0
votes
1answer
38 views

What is the value of the measure of a line segment?

Let $$f(x)=1-x^2$$ Then $$|\{x\in\mathbb{R^1}:f(x)>0\}|=|(-1, +1)| = 2$$ Let $f$ be a nonnegative function, defined on measurable subset $E$ of $\mathbb{R}^n$. Then $\Gamma(f, ...
1
vote
1answer
18 views

Establishing set theoretic identities

I want to prove that $$ \bigcup_{A \in \mathcal{P}(X)} A = X, \; \text{and} \; \bigcap_{A \in \mathcal{P}(X)} A = \varnothing $$ Attempt: For the first one, If $Y \in \bigcup_{A \in ...
1
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0answers
29 views

Lower bound on the difference between max. and min. values of a polynomial over $[-1, 1]$

Problem: $P(x)$ be a monic, n-degree polynomial with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(x) < \frac{1}{2^n}$$. I tried it to put ...
2
votes
0answers
19 views

Variation of Steinhaus Theorem

In the proof of the problem, For $A$ and $B$ measurable subsets of $\mathbb{R}$ with finite measure, show that $A+B$ contains an interval of positive length, it says that ...