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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36 views

When a limsup can enter inside an integral

In the book "partial differential equation in classical mathematical physics" by I.Rubinstein,L.Rubinstein at page 411 I found something that I can't justify. It seems that the author (between ...
5
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1answer
71 views

Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
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1answer
58 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but ...
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42 views

Dual of a maximization problem

We have a positive, smooth, increasing concave function $f:\mathbf{R}^n\to \mathbf{R}^+$ and $k$ smooth, increasing constraint functions $f_i:\mathbf{R}^n\to\mathbf{R}$. I've recently encountered two ...
1
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1answer
41 views

Approximation of $\sin\left(\dfrac{x}{n}\right)$

I wrote in my analysis notes the following: $\sin\left(\dfrac{x}{n}\right) = -\dfrac{x}{n} + \omicron\left(\left| \dfrac{x}{n} \right|\right)$. I'm guessing it comes from Taylor's formula but I ...
2
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2answers
83 views

Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
8
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52 views

How do I prove that $f_n\to f$ in $L^p$?

Let $\{f_n\}$ be a sequence in $L^p([0,1])$ for $p\geq 1$. Suppose there exists $f\in L^p([0,1])$ satisfying $\lim_{n\to\infty} \int_0^1 f_n(x)g(x)dx = \int_0^1 f(x)g(x)dx$ for any $g\in L^2([0,1])$. ...
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2answers
39 views

About converging sequence of norm-one elements of $\ell^2$

I got stuck in this question for days, so any help/hint is appreciated. Assume that $T$ is the unit ball in $\ell^2$ and $(x_n)_{n=1}^{\infty} \subset T$ and that $(x_n(i))_{n=1}^{\infty}$ converges ...
5
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2answers
87 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
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0answers
14 views

Function crossings

Determine the number of positive real roots of $a^x = x^a$, ($a>0$). For $a=2$ there are $2$. For other positive integers other than $a$ there's $1$. I suspect there's range of $a$ for which ...
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1answer
27 views

$A$ is uncountable and $B$ divides $A$ in two uncountable sets. Show that $B$ is nonempty and open

This is the exercise 3.2.12 of Understanding analysis 2ed. of Abbott: Let $A$ be an uncountable set and let $B$ be the set of real numbers that divides $A$ into two uncountable sets; that is, ...
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0answers
12 views

Should we expect any anomalies when dealing with rational differential equations?

Some time ago, I found myself reading a short article that proposed that the rational numbers where the 'appropiate' number system that we should use for most of mathematics (where we use the real ...
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0answers
18 views

sard's theorem proof step by step

Hello guys I want to ask your help. I have tried to understand the proof of the sard's theorem because I have to present it in a final presentation that I have to make in the course of vectorial ...
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2answers
29 views

As $x \to 0$ this approaches zero since the zero of $e^{\frac{-1}{x^2}}$ beats out the pole of $\frac{1}{Q_k(x)}$?

In the Robert Strichartz's book "A Guide to Distribution Theory and Fourier Transforms" at the page $4$, we have an interesting exercise : $$ \psi(x) = \begin{cases} e^{\frac{-1}{x^2}} ...
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0answers
16 views

Derivative of supremum (if exists) bounded by supremum of derivatives?

Recently I came across the following problem: Suppose that $\{g_\alpha\}$ is a class of real differentiable functions on some interval $I\subset \mathbb{R}$, and $(\sup g_\alpha)'(t)$ exists for some ...
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0answers
18 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
2
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3answers
89 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
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0answers
23 views

Uniform convergence of a series on the real line

Let $f_n(x)=e^{-(x-n)^2}$ and let $ g(x) = \begin{cases} \frac{1-e^{-x^2}}{x^2} & x \ne0 \\ 1 & x=0 \end{cases}$ Suppose $g$ is continious, bounded and have maximum at 0 Show that ...
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3answers
21 views

If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N\to f_n(X)\subseteq U$.

Let $X$ be a compact, $U$ open set and $f:X\to\mathbb{R}$ continuous such that $f(X)\subseteq U$. If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N$ ...
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1answer
38 views

How do I show that a contraction mapping in a metric space is continuous?

I start out by letting $V$ be an arbitrary open set in $X$. Then $$ f^{-1}(V) = \{x\in X\mid f(x) \in B_\epsilon(f(a))\}. $$ This can be re-written as: $$ f^{-1}(V) = \{x\in X\mid d(f(a), f(x)) < ...
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22 views

About the proof of a corollary of Arzela-Ascoli Theorem.

This is from Scheidemann, Complex Analysis. Theorem (Arzela-Ascoli): Let $K$ be a compact separable metric space, $E$ a finite-dimensional Banach space and $(f_j)_{j\in\mathbb{N}}\subseteq C(K,E)$ ...
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2answers
43 views

$(f_n) $ converge uniformly on $X$.

Let $(f_n)$ be a sequence function continuous $f_n:X\to \mathbb{R}$ that converge uniformly on $D\subseteq X$ dense set. Then $(f_n) $ converge uniformly on $X$. can someone help me with this. If X ...
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0answers
12 views

Why may points of jump have finite point of accumulation? [on hold]

Specifically, what does it mean by finite point of accumulation? How is the concept connected with jump?
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3answers
40 views

Is there any way to calculate the roots of this polynom?

I need to calculate the roots of the real function $f$: $$ f(x)=\frac{-{x}^{3}+2{x}^{2}+4}{{x}^{2}} $$ But I am not able to decompose the numerator. There should be only one real solution and two ...
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3answers
105 views

If a Riemann integrable function is zero on a dense set, then its integral is zero

Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function such that $g(x)=0$ for all $x\in A\subseteq[a,b]$ where $A$ is dense set. Then $$\int_{a}^{b} g=0$$ How can I show this?
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4answers
35 views

Relation between counting measure and Tonelli theorem

This is from Rudin's RCA book. But I can't understand how he got Corollary. What he takes as $f_n, X$? If we consider counting measure how integral converts to sum? I can't show this after some ...
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32 views

$f$ is $3-$times differentiable and has at least $5$ distinct real zeroes, prove $f+6f'+12f''+8f'''$ has at least two distinct real zeroes?

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeroes. Prove that $f+6f'+12f''+8f'''$ has at least two ...
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1answer
42 views

Question about Lebesgue integral

When I prove a real analysis problem, I need a theorem about Lebesgue integral, but I cannot find this theorem in any standard reference, intuitively, I think it is correct, but I do not know how to ...
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0answers
27 views

Order Size estimation of converging sum used for approximation of logarithm

I know it can be shown that $\log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i}$ for $\forall n\in\Re$ where $n\ge1$ For given natural m, I tried to find the order size of k = f(m,n) in order for the ...
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1answer
16 views

Closure of set intersection neighborhood with the closure set

What's the meaning of the definition of the closure set that if (x belongs to A ) the neighborhood of x intersects with A does not empty I'm trying to prove that the closure of A is the smallest ...
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1answer
48 views

If $\int_{a}^{b}f=0$ then $\{x\in[a,b]:f(x)=0\}$ is dense set.

Let $f:[a,b]\to\mathbb{R}$ be a function Riemann-integrable such that $f(x)\geq 0$ for all $x\in[a,b]$. If $\int_{a}^{b}f=0$ then $\{x\in[a,b]:f(x)=0\}$ is dense set. Can someone help me with this ...
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0answers
19 views

If $\{x\in[a,b]:f'(x)=0\}$ be a null set then $f$ is increasing

Let $f:[a,b]\to\mathbb{R}$ be a function differentiable such that $f'$ is Riemann-integrable and $f(x)\geq 0$ for all $x\in[a,b]$. Question: If $\{x\in[a,b]:f'(x)=0\}$ be a null set then $f$ is ...
2
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2answers
23 views

My personal way to try to show that if $L$ is the set of limits points of $A\subset\Bbb R$ then $L$ is closed

Im trying to make the proof on this way: I can define the set of limits points of $A$ as $$L=\{l:(a_n)\to l, a_n\in A\}$$ The hypothesis is that $L$ is closed, i.e. $\forall (l_n)\to m\implies m\in ...
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22 views

If $\int_{a}^{b}fg=f(a)\int_{a}^{b}g$ there is $c\in(a,b)$ such that $f(a)=g(c)$ [on hold]

$f:[a,b]\to \mathbb{R}$ be a continuous function and $g$ is Riemann-integrable such that $g(x)\leq 0$ $\forall x\in[a,b]$. If $\int_{a}^{b}fg=f(a)\int_{a}^{b}g$ there is $c\in(a,b)$ such that ...
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1answer
30 views

$L_p(\mu)\subseteq L_q(\mu)$ [on hold]

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?
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0answers
36 views

$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
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42 views

axiom of continuity guarantees no gaps exist on the real axis?

I read this content at the bottom of this page just wonder why axiom of continuity could guarantee no gaps exist on the real axis? any proofs ?
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2answers
36 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
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14 views

Compact set of function involving BSpline functions.

Let $X = \left\{x_0,...,x_{n-1}\right\}$, $x_i-x_{i-1} = h$ for $i=1\ldots n-1$ and $$ \phi(x;X,\vec{\beta}) = \sum_{j=0}^n \beta_j \phi_{j,2}(x;X) $$ where $\phi_{j,2}$ is the $j$-th BSpline (of ...
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1answer
44 views

Intuition behind proving that $\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$

Okay, so I'm having a bit of an issue understanding Rosenlicht's proof that $$\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$$ I've worked out the proof and I have written it down myself, but ...
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28 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
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31 views

Name of dominated convergence for sums

Having a sequence $(a_n(j))_{n}$ where every element of the sequence also depends on $j\in\mathbb{N}$. If $\sum_{n=1}^\infty \sup_{j\in\mathbb{N}} |a_n(j)| < \infty$, then the following (assuming ...
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1answer
60 views

Prove uniform convergence of series

I'm given to functions, $f_{n}(x)=e^{-(x-n)^2}$ and $g(x)= \begin{cases} \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ 1 & x=0 \end{cases}$. I have to prove that $$\sum_{n=0}^{\infty} g(x) \cdot f_n(x) ...
3
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1answer
21 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
3
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1answer
28 views

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

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 ...
5
votes
2answers
479 views

dy/dx=0 has uncountable many solutions

Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible?
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0answers
20 views

Uniform convergence of sum in given interval

Let $a>0$ and observe $ln(a) + \sum_{n=1}^{\infty} -\frac{1}{n}a^{-n}x^n$ Show that the above sum converges uniformly in $[-a,b]$ with $-a<b<a$ Here's my current approach: For $x \in ...
1
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1answer
51 views

Dimensions of a sphere and a ball

The volume of the unit ball in $\mathbb{R}^n$ is denoted by $v_{n}$ and the surface area of the unit sphere $S^{n-1}$ is denoted by $\omega_{n-1}$. What is the importance of writing $n-1$ and $n$?
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0answers
13 views

Examples 3.35 (a) and (b) in Baby Rudin: Limit Superior and limit inferior of a couple of sequences

This question is related to Examples 3.35 (a) and (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, p. 67. Let us consider the series $$ \frac 1 2 + \frac 1 3 + ...
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0answers
15 views

Establishing a set identity involving collections

$X \cup ( \bigcap_{A \in \mathscr{A}} A ) = \bigcap_{A \in \mathscr{A} } (X \cup A ) $. Attempt We have $x \in X \cup ( \bigcap_{A \in \mathscr{A}} A )$ iff $x \in X$ or $x \in A $ for all $A ...