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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4 views

On lower envelope of a map

Let $X$ be a metric space and let $f:X\to\mathbb R$. The lower envelope of $f$ is $g(x)=\sup_n\inf_{y\in B_{1/n}(x)}f(y)$, where $B_\epsilon(x)$ is the ball centered at $x$ with radius $\epsilon$. ...
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4 views

Limit points of a Set Question

Suppose we have the set , {(x,sin(1/x): with x contained between 0(not included) and 1/pi (included). I have to find the Closure of this set and to this, I need to consider its limit points. Can ...
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3answers
58 views

Show $\int_0^1 \frac{\log(1-x)}{x}=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
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1answer
16 views

Suppose $(X,d)$ is a metric space. Let $p$ belong to $X$. What can you mention about the connectivity of $X$, if $\{p\}$ is open?

Suppose $(X,d)$ is a metric space. Let $p$ belong to $X$. What can you mention about the connectivity of $X$, if $\{p\}$ is open?
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1answer
58 views
+50

Why this set is dense in $C_0(\mathbb{R})$

Let $C_0=\{f~|~ f:\mathbb{R}\to\mathbb{R},f~is~continous,\lim\limits_{\vert x\vert \to\infty}f(x)=0\}$ $A=\{f~|~f(x)=p(x)e^{-x^2},p(x)~is~polynomials\}$ Why $A$ is dense in $C_0$. The topology ...
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0answers
31 views

Part of proof of term-by-term integration

I want to prove the theorem of term-by-term integration for lebesgue integrable functions (denoted as $L^1$ functions): Suppose $(g_n)$ is a sequence of $L^1$ functions over a measure space $(X,\sigma ...
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1answer
5 views

Closed set with finite measure

Assume $F\subset \mathbb R$ is closed and $\mu(F)<\infty$, where $\mu$ is the Lebesgue measure. Can we deduce that $F$ is bounded?
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1answer
47 views

Real Analysis Qualifying Exam Practice Questions

I'm doing some practice questions for a real analysis qualifying exam coming up in a few weeks. I have a couple questions, namely on the "if the statement is true, prove it. Otherwise, give a ...
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3answers
18 views

Ordinals and Countability

I've been watching Prof Su's (Harvey Mudd College) incredible lecture on Ordinals and Transfinite Induction, and I have a few related questions. He starts by drawing and examining three sets of dots, ...
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2answers
30 views

Convergent subsequence in a bounded sequence of a complete metric space

Consider a complete metric space E with the following property: If $x_n$ is a bounded sequence, then $\forall \epsilon > 0$, $\exists i,j , i \neq j$ such that $d(x_i,x_j) < \epsilon$. ...
3
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2answers
24 views

Chain Rule and Vector valued functions?

Let $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just ...
4
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1answer
62 views

The sequence $(-1)^n\binom{\alpha-1}{n}$ converges.

I need to show that for $n \in \mathbb N_0$ and $\alpha \ge 0$ the sequence $(-1)^n\binom{\alpha-1}{n}$ converges. It can be shown that the sequence convereges to zero using a theorem claiming that ...
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1answer
83 views
+100

(Elegant) proof of an inequality: $h(x) \geq 1- (1-\frac{x}{1-x})^2$

I am looking for the most concise and elegant proof of the following inequality: $$ h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1) $$ where $h(x) = x \log_2\frac{1}{x}+(1-x) ...
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0answers
14 views

Intergration $\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$

I need to calculate the integral: $$\int^{\infty}_{0}\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$$ I guess complex analysis is suitable for this integral, but I still have no ideas which kinds of trick ...
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1answer
390 views

Proof of the Cauchy Criterion for Series

I was trying to come up with my own very simple proof of the Cauchy Criterion for Series $\left(\text{i.e., let}\, \{a_{n}\}\, \text{be a sequence in}\, \mathbb{R}. \text{The ...
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3answers
140 views

Uses of step functions

My highschool teacher has informally told us about what continuity is and used step functions as an example of a discontinuous function. The Wikipedia page for it links to a lot of other kind of step ...
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1answer
42 views

Simply connectedness of spherical shell

Consider a spherical shell $U$ in $\mathbb{R}^3$(the open region between two spheres). I want to show that any closed curve in $U$ can be shrunk into a single point without leaving $U$. This exercise ...
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2answers
22 views

joint infima involved in relation to distance between sets

Consider a metric space $(X,d)$, and let $A,K \subseteq X$ such that A is closed and K is compact. I have to show that there exists an element $k_0\in K$ which achieves the minimum between the sets, ...
2
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1answer
28 views

If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

Let $a_{n} \geq 0$ and $b_{n}>0$ for each $n$ in $\mathbb{N}$ and suppose that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} ...
2
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0answers
32 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
2
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1answer
19 views

show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
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1answer
40 views

Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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2answers
27 views

Prove that there exists some real number θ satisfying 0 < θ < 1 for which f '''(θ) = 0

Let f: D → R be a 3-times differentiable function defined over an open interval D, where 0 ∈ D and 1 ∈ D. Suppose that f(0) = f '(0) = 0 and f(1) = f '(1) = 0. Prove that there exists some real ...
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3answers
134 views

$f>0$ on real line ; $f(x+y)\le f(x)f(y) , \forall x,y \in \mathbb R$ ; $f([0,1])$ is bounded set ; does $\lim_{x \to \infty}(f(x))^{1/x}$ exist?

Let $f: \mathbb R \to (0,\infty)$ be a function such that $f(x+y)\le f(x)f(y) , \forall x,y \in \mathbb R$ and $f$ is bounded on $[0,1]$ ; then does the limit $\lim_{x \to \infty}(f(x))^{1/x}$ exists ...
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1answer
16 views

Suppose that f: [1,2] -> [0,3] is continuous. Show that there is some t exists in [1,2] such that f(t) +3 = 3t

So what I did was: f(t) +3 = 3t f(t) = 3t - 3 f(t) = 3(t-1) then substituting in 1 and 2 to get 0 and 3. then by IVT there exists t in [1,2] such that f(t) - 3 = 3t that proves the [0,3].
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1answer
93 views

For any measure right?

Let $ A , B \subset \mathbb{R}$. Does $m^* (A \cup B) + m^* (A \cap B) \le m^* (A) +m^* (B)$ hold for every measure? Example, for outer measure, Lebesgue measure, etc.
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1answer
21 views

$S$-differentiable functions.

I just thought of this fun exercise: Let $S=\bigcup \{D\subseteq \Bbb R: \text{$D$ is dense in $\Bbb R$ and $|D|=\aleph_0$}\}$. We say a function $f:\Bbb R\to \Bbb R$ is $S$-differentiable at $x$, ...
3
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1answer
39 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then ...
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0answers
19 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
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0answers
49 views
+50

Which properties characterize $\sin, \cos$?

I know a few properties of $\sin$ and $\cos$, for example: $\sin^2+\cos^2=1$ $\sin (a+b) = \sin a\cos b+\cos a\sin b$. $\cos (a+b) = \cos a\cos b-\sin a\sin b$. $\sin (x+\delta) = \sin x$ for some ...
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0answers
10 views

Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
3
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2answers
116 views

What are “set-theoretic maps”? [closed]

Can someone explain to me what is the meaning of “set-theoretic maps”? I've encountered this term in real analysis in $n$ variables. Specifically, I encountered it in the following statement: Let ...
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1answer
36 views

Set is Unbounded

Let $S$ be a nonempty set and $U$ be an open set contained in $S$. Let $C$ be a closed set such that $S\subseteq C$. Assume that $C\setminus U=\emptyset$. Prove that $S$ is unbounded. I have that ...
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1answer
24 views

Questions on multidimensional integrals

$$ f(x,y) = \begin{cases} x^2(1 − y)^2 & 0 ≤ x ≤ y ≤ 1 \\ x + y & 0 ≤ y < x ≤ 1 \end{cases} $$ Argue that $f$ is continuous at all points in $[0, 1] × [0, 1]$ that are not in the closed ...
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4answers
10k views

An inflection point where the second derivative doesn't exist?

A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points ...
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1answer
17 views

About a measurable set

Let $A$ be a Lebesgue measurable set with $m(A)=8$. How to show that there exists a measurable set $B\subset A$ such that $m(B)=5$? I am not getting any idea how to proceed? Any hint will be ...
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0answers
14 views

limsup and liminf of a function explenation

I have a little problem at understanding limsup and liminf of a function at a point y or infinity..I would like to understand the definitions and the theorems formally and intuitively..Can someone ...
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1answer
58 views

How to calculate $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$

Problem Show that $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$ by the following consequence. Consider the sequence $(s_n)$ defined by $s_n=1 + 1/2 + \cdots + 1/n$. ...
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1answer
34 views

$\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
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5answers
93 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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0answers
23 views

Question about n- expansive homeomorphism

Let $(X, d)$ be a compct metic space and $f$ be a homeomorphism on $X$ . Suppose $\Gamma_c(x)=\{y: d(f^{n}(x), f^{n}(y))<c \ , \forall n\in Z\}$ and for some $z\neq x$, $z\in \Gamma_c(x)$. ...
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0answers
33 views

Proving that the limit is zero

Can someone please help me to answer the following question: We consider a function $u\in L^2(R^d) $ which satisfies the following inequality: $\|R(x)u\|^2 \le \|u\|^2$ By using a partition of ...
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Baby rudin exercise 2.16 [on hold]

Click here for question and solution This is a problem from baby rudin exercise 2.17.I have attached the solution.I can't understand how that 7/9 arises in that inequality instead of it just being 7. ...
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1answer
15 views

Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces satisfying the Heine-Borel theorem. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that ...
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1answer
19 views

Existence of Solution to Integral Equation

How do I show that the integral equation \begin{equation*} x(t) = \ln(1+t) + 1/2\int_0^1e^{-t}\sin^2(ts)x(s)ds \end{equation*} has a solution $C[0,1]$?
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0answers
10 views

reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
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1answer
126 views

Simple example of uniformly continuous and continuous

I'm struggling somewhat with the difference between 'uniformly continuous' and 'continuous'. I understand the intuition, but I need to see it in an example. Say $f=2x$, or some other easy function ...
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0answers
12 views

Convergence of Iterative First Order ODE Method

Let us suppose we have a function $f(x,t)$ which is Lipschitz wrt $x$. Then we have the following iterative method $$ x_{m+1}(t) = c+\int_{t_0}^tf(x_m(v),v)dv$$ for solving the ODE $x'(t) = ...
1
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0answers
19 views

How to find out the the minimum value of the given integral?

What is the value of $min_{f\varepsilon D} \int_{0}^ {1} (1+x^2)f^2(x)dx$ where $D$={$f:[0,1] \to \mathbb R : f continuous, \int_{0}^{1} f(x)dx =1$} I have no idea how to look for minimum value? ...
0
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0answers
39 views
+100

Regularity, Dirichlet form

I have a question about Dirichlet form. Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and $X=\bar{\Omega}$. The measure $m$ on the Borel $\sigma$ algebra $\mathcal{B}(X)$ is given by ...