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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11 views

How can we prove $\int_{B_\rho(x_0)}\Delta u\;d\lambda_n=\int_{\partial B_1(0)}\frac{\partial u}{\partial\rho}(x_0+\rho\omega)\rho^{n-1}\;d\omega$?

Let $B_r(x_0)$ and $\overline{B}_r(x_0)$ be the open and closed ball in $\mathbb{R}^n$, respectively $u\in C^2(B_r)$ and $\rho\in (0,r)$ $\lambda_n$ be the Lebesgue-measure on the ...
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1answer
32 views

Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
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2answers
17 views

An open and closed ball in the discrete space

let $(X,d)$ be a metric space. I am trying to find what an open and closed ball looks like in the discrete space, i.e. when $d(x,y) = 0$ for $x = y$ and $1$ otherwise. Just considering the open ball ...
3
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3answers
36 views

How to evaluate $\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx\:,\:n\in \mathbb{N}$ First step I wrote that $\int ...
3
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1answer
17 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
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10 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
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1answer
14 views

Fourier transform of Schwartz functions

I am stuck with the following question: Suppose $f \in \mathcal{S}(\mathbb{R})$ satisfies $\widehat{f}(\xi)=0$ for $|\xi|<1.$ Prove that there exists $g \in \mathcal{S}(\mathbb{R})$ such that ...
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1answer
12 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
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1answer
14 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
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1answer
15 views

What kind of information is available in a Fourier series expansion of an analytic function that is not (readily) available in a Taylor series?

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?
2
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1answer
35 views

Show that this path is differentiable but not rectifiable

My path is defined as follows: $\gamma:[1,1]\rightarrow \mathbb R, \space \gamma(t):= \begin{cases} \ (0,0) & \text{if $t$=0} \\[2ex] t,t^2 \cos (\frac{\pi}{t^2}), & \text{if $t$ $\in$ ...
3
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1answer
32 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't place nicely with ...
2
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0answers
21 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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0answers
13 views

Doubt on asymptotics of continous functions (little-o notation and taylor expansion).

Suppose I have $e^{(\frac{1}{n}b + o(\frac{1}{n}))}$ then $\lim_{n \rightarrow \infty} = e^0 = 1$ so $$e^{(\frac{1}{n}b + o(\frac{1}{n}))} = o(1) +1$$ But if I take the Taylor expansion of ...
2
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0answers
27 views

Too strong assumption in the Uniqueness Theorem of Rudin's Real and Complex Analysis?

In Rudin's Real and Complex Analysis, there is the following result about Fourier transforms. The Uniqueness Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}(t)=0$ for all $t\in\mathbb{R}$, then ...
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2answers
19 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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0answers
11 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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1answer
27 views

finding the continuity of a function

I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where ...
3
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4answers
247 views

Why Riemann sum is less than value of the integral?

Why $ \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}\leq\int_{0}^{1}\frac{dx}{1+x}=\log 2 $ ? because I think: $$\int _0^1\frac{dx}{1+x}=\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}}$$ Why Riemann ...
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0answers
23 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
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1answer
13 views

Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
6
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5answers
228 views

Why the radius of convergence and not “areas of convergence” for power series?

My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence. I am comfortable with ...
3
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7answers
372 views

Error in my proof?

What is wrong in this proof. It seems correct to me but still doesn't make proper sense. $=\sqrt{...\sqrt{\sqrt{\sqrt{5}}}}$ $=5^{1/\infty}$ $=5^0$ $=1$ EDIT So does this mean that ...
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1answer
23 views

double root and newton method, a problem on solved exercise?

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
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2answers
36 views

How we can prove that: $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot log\left(2\right)$?

$f:\left[0,1\right]\rightarrow R,\:f\left(x\right)=\frac{1}{1+x}$ and we have to show that $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot\log\left(2\right)$.What I know is just that: ...
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1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
0
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1answer
29 views

a continuous function

Let $C([a,b])$ be the collection of all functions $f:\mathbb{R} \to \mathbb{R}$ such that continuous on $[a,b]$. It is known that if $f\in C([a,b])$ then $f$ is continuous on every sub-interval of ...
3
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2answers
104 views

maximum area of semi-circle in square

I'm struggling the with the following question: Given is a square with length $a$. Now I want to find a semi-circle with the max. area. Looks like this: ...
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2answers
67 views

A theoretic question about cosine general solution.

I have to find the extremas of: $f(x)=x-\tan({x\over 2})$ .$(\pi\le x\le\pi)$ Last result is $\cos({x\over 2})=\pm{1\over \sqrt{2}}$. I get that: ${x\over 2}=\pm{\pi\over 4}+2\pi k$ which derives: ...
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3answers
89 views

Proving $\left(a+\frac{2}{a}\right)^2+\left(b+\frac{2}{b}\right)^2\ge \frac{81}{2}$ for all positive real $a,b$ such that $a+b=1$

I approached this problem in two different ways, but only one was successful. I'll post the latter as an answer, while here follows the first approach: I expanded the squares: ...
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0answers
15 views

Isn't $\gamma_{n,g}=0$ if $g>n-3$?

Let $\gamma_{n,g}$ the number of ways of putting down x $y's$ on the intervall $[0,n-1]$ with the $y's$ separated by at least two $z's$ and let $\gamma_n=\sum_{g=0}^{n}\gamma_{n,g}$. Maybe a stupid ...
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0answers
6 views

References for a notion related to radially lower semicontinuity

Let $E$ be a real vector space, $C\subset E$ be a nonempty convex set and $z\in C$. Let $f:C\rightarrow\mathbb{R}$ such that $$ \textbf{(A)} \quad f(z)\leq\limsup_{t\downarrow 0}f(z+t(w-z))\quad ...
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1answer
12 views

Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
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0answers
14 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
1
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1answer
22 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
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2answers
35 views

Use the Mean Value Theorem to prove [on hold]

Use the Mean Value Theorem to prove that $|\sin x - \sin y| \leq |x - y|$ for all $x,y \in \mathbb{R}$.
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1answer
49 views

Rational points on circle

I need help for the following questions. Give the necessary and sufficient condition for $r$ such that the circle $x^{2}+y^{2}=r^{2}$ passes the rational points. I know the obvious sufficient ...
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1answer
54 views

Proving $\lim_{x\to 0}f(x)=\infty$.

Suppose we have a function $f:[0,\infty)\to \mathbb{R}$ such that for every $N\in\Bbb{N}$ and every sequence of $\delta_n>0$ such that $\lim_{n\to\infty}\delta_n=0$, there exists $n$ for which ...
5
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2answers
72 views

computing an integration with a floor function

I am trying to compute $$\int_0^1 \left(\frac{1}{x} - \biggl\lfloor \frac{1}{x}\biggr\rfloor\right) dx$$ with no success. Any hints?
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1answer
38 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
0
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1answer
32 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable? [on hold]

If the answer is yes, how to prove that? Otherwise how to find a counterexample?
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0answers
35 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
0
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1answer
53 views

Is an everywhere differentiable function locally Lipschitz?

If we have a differentiable function $ f:\mathbb{R}^n \to \mathbb{R}^n $, does it have to be locally Lipschitz? It's obviously true for continuously differentiable functions, but what happens without ...
4
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1answer
31 views

Difficult exercise on unicity of solutions for an IVP

Suppose $f$ and $g$ are continuous and $g$ is odd and strictly increasing function. I have to prove that the IVP $$y'=f(x)g(y)$$ $$y(0)=1$$ has a unique solution if and only if $$\lim \limits_{u \to ...
1
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1answer
14 views

Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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3answers
64 views

Implicit Function Theorem Zero

The Implicit Function Theorem gives conditions under which f(p)=0 can be solved for some variables in terms of the rest. One of the conditions is f(p0)=0. How do you know a p0 exists and how do you ...
2
votes
1answer
58 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
2
votes
1answer
40 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
0
votes
1answer
26 views

Pointwise Limit of $f_n = \frac{x^n}{2+x^{2n}}$.

Let $f_n = \frac{x^n}{2+x^{2n}}$. What is the pointwise limit of this sequence of functions on $(0,1)$? We cannot say anything about $\frac{1}{x^n}$ since $|x| < 1$.
0
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1answer
54 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...