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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1answer
17 views

If a continously differentiable function has a local minimizer, can it be one to one?

Let $f$ be a continuously differentiable function defined $f : \mathbb R \to \mathbb R$ such that $f(x)$ is defined for for all $x$. Suppose $x_0$ is a local minimizer for $f$. Is $f$ one-to-one? I ...
1
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1answer
28 views

Convergence of Taylor Series

Prove that if $f$ is defined for $|x|< r$ and if there exists a constant $B$ such that $$| f^n(x) |\le B$$ for all $|x|< r$ and $n \in \mathbb N$, then the Taylor series expansion : ...
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2answers
40 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...
1
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0answers
13 views

Kernel function related proof

This is not my area of expertise -- nonetheless, I need some sort of semi-convincing proof of the following equation, which has been cited in several machine learning articles I've read: $$ d_j = ...
1
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2answers
29 views

Prove that $F=\left(-x^2+2\right)\cdot \cos\left(x\right)+2x\sin\left(x\right)$ don't have limit

We have $f\colon\mathbb{R}\rightarrow \mathbb{R}$, $f\left(x\right)=x^2\cdot \sin\left(x\right)$ and $F$ its primitive. We have to prove that $F$ doesn't have a limit at $\infty $. What I can say ...
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0answers
23 views

Approximating simple function by continuous function

I am trying to solve this problem: If $\gamma$ is a simple function defined on $E \subset \mathbb R^d$, $E$ measurable, then there is $f:E \to \mathbb R$ continuous such that $$|\{x \in E: f(x) \neq ...
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1answer
29 views

How can I prove that $\int_X\left(\int_Y f_xdm_2\right)dm_1$ exists given the following conditions …?

Let $X=Y=[0,1)$ and $f(x,y)=\dfrac{1}{(1-xy)^a}$, where $a>0$, and $m_1=m_2$ the Lebesgue measure. I want to prove that $$\displaystyle\int_X\left(\int_Y f_xdm_2\right)dm_1$$ exists (the integral ...
1
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2answers
29 views

How can I make a series expansion of $F(x) = \int_0^x \exp -{(t^2)}\ dt$?

$$F(x) = \int_0^x \exp -{(t^2)}\ dt$$ We need to find the series expansion for $F(x)$. I tried differentiating $F(x)$ but couldn't establish certain pattern so that Taylor series formation may help.. ...
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0answers
16 views

Jordan region problem

What is a Jordan Region? I can't find the definition anywhere. The question asked, if $E\subset \mathbb{R}^n$, bounded and with finitely many accumulation points, then $E$ is Jordan region.
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18 views

a connect set in plane [on hold]

Let $$E=\{(x,y)\in\Bbb{R^2}:x\in\Bbb{Q^c}\,\ or\,\,y\in\Bbb{Q^c}\}$$ show that $E$ is connect,is $$E=\{(x,y)\in\Bbb{R^2}:x\in\Bbb{Q}\,\ or\,\,y\in\Bbb{Q}\}$$ connect?
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3answers
24 views

Set that is bounded but not totally bounded: Reading textbook

I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally ...
0
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1answer
20 views

Why is just $0$ extreme point? v22

We have $f:R\rightarrow R,\:f\left(x\right)=x^3-3x+2$ and we need to find extreme points for $g:R\rightarrow R\:,\:g\left(x\right)=\int _0^{x^2}\:f\left(t\right)e^tdt$. Here is all my steps: ...
-2
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3answers
29 views

A problem of Schwarz derivative

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
5
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1answer
30 views

Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
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0answers
23 views

fn converges to f pointwise where all functions fn are bounded and f is unbounded: Is this example correct?

I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded. I have thought of an example where the ...
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0answers
36 views

Which is strongest of all? [on hold]

lipschitz condition, uniformly continuous, differentiable and which is weakest? please help
1
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1answer
17 views

Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
2
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0answers
24 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
2
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1answer
26 views

How can I find monotonicity intervals? v18

We have $F:\mathbb{R}\rightarrow \mathbb{R}$, $F(x)=x\int _0^x (1+\cos(t)) \, dt$ and we neeed to find monotonicity intervals and I don't know how... Here is what I try to do: $$F'(x)=\int _0^x ...
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0answers
21 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
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0answers
33 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
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1answer
16 views

What is the outer measure of the union of uncountably many sets of measure 0.

I know that the union of countably many sets of measure 0 has measure 0. How about the case of uncountably many of them?
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0answers
14 views

the Fisher equation has no positive traveling wave solution

Use the linearization method to prove that for any $c\in(0,2)$,the Fisher equation$u_t=u_{xx}+u(1-u)$has no positive traveling wave solution $U(x+ct)$ with $U(-\infty)=0$
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2answers
20 views

How we can prove that $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n f(t)\:dt$ is convergent?

We have $f:\left(-1,\infty \right)\:\rightarrow \:R,\:f\left(x\right)=\frac{x}{x+1}$ and we need to prove that: $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n\:f\left(x\right)dx$ is convergent.Maybe, in ...
0
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0answers
13 views

Lebesgue measure of union of disjoint measurable sets

I am wondering if the Lebesgue measure of the union of a countable collection of disjoint measurable sets is equal to the sum of the measure of such sets. I feel that they may be equal. I know that ...
1
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3answers
34 views

Prove that this set is open

$$ A=\left\{ x \in \mathbb{R}^{p} \mid \forall i:\ x_{i} \in (-1,1) \right\}$$ Pick $x\in A$ at random, and choose $\delta = \min B$ where: $$ B = \{ 1-x_{i}\mid i \in \{1,2,\dotsc,p\} \} \cup \{ ...
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4answers
45 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
0
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1answer
14 views

Continuity of $f$ on $I$ where $f_n$ is continuous on $I$ and it converges uniformly to $f$ on $I$

Let $I=[a,b]$ be a bounded and closed interval, let $f_n$ be a sequence of functions on $I$ and $f:I\rightarrow\Bbb{R}$. $f_n$ is continuous on $I$ for all $n\in\Bbb{N}$ and it converges uniformly to ...
0
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2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
0
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1answer
22 views

Evaluate $\lim _{n\to \infty }\left(1-f\left(\frac{1}{\sqrt{n}}\right)\right)\cdot \sum _{k=1}^nf\left(\frac{k}{n}\right)$

We have $f:\left[0,\frac{\pi }{2}\right]\rightarrow R,\:f\left(x\right)=cos\left(x\right)$, and we need to evaluate: $\lim _{n\to \infty ...
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0answers
7 views

Kernel estimate in boundary point

Good moorning, I wonder how to prove that if $X_{1}, \ldots, X_{n}$ are iid from exponential distribution with expected value 1, then the expected value of its kernel density estimator in zero is ...
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1answer
25 views

Translation property in $L^1(\mathbb{R})$ space

Let $g(x)$ be a bounded measurable functions on $\mathbb{R}$, and $f(x)$ be in $L^1(\mathbb{R})$. Notation: $\int_\mathbb{R} h(x)dx=\ $the integration of measurable function $h$ over $\mathbb{R}$ I ...
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2answers
46 views

How do I show that $0<a_n^2<a_n$ If $\sum _{n=1}^\infty a_n$ is convergent?

Since $\sum _{n=1}^\infty a_n$ converges, i know know that $\lim _{n\to \infty}a_n=0$. I know I have to use the comparison test to show that ${a_n}^2$ converges but how do i show that ...
4
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1answer
88 views

Where is Cauchy's wrong proof?

Allegedly, Cauchy mistakingly "proved" that pointwise convergence of continuous functions is continuous. I saw this somewhere in a book, and it is also in wikipedia: Uniform convergence. In his ...
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0answers
31 views

Shifting to polar coordinates

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in ...
0
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2answers
43 views

Power Series Problem… [on hold]

If $$\sum_{n=0}^\infty a_n x^n=0$$ does that imply $a_n$ are equal to zero?
2
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1answer
33 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
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1answer
31 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
0
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1answer
28 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
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1answer
34 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
0
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1answer
24 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
2
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0answers
48 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
1
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2answers
41 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
1
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1answer
30 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
1
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2answers
38 views

Prove that $\int_0^1\int_x^1 \frac{f(y)}ydy\,dx=\int_0^1f(x)\,dx$ if $f$ is Lebesgue integrable

Suppose f is a Lebesgue integrable function on [0,1] and define a new function by $$g(x)= \int_x^1 \frac{f(y)}ydy$$ for all x in [0,1]. Prove that $$\int_0^1{g(x)} dx=\int_0^1{f(x)} dx$$ My ...
1
vote
1answer
28 views

Prove that a set is sigma finite

Let $(X, \mathcal{F}, \mu)$ be a $\sigma$-finite measure space and let $A$ in $F$ be such that $\mu(A)=\infty$. Prove that there exists $B$ in $ \mathcal{F}$, $B\subset A$, such that $0< \mu(B)< ...
3
votes
1answer
15 views

Problem with real differentiable function involving both Mean Value Theorem and Intermediate Value Theorem

Problem: Let $a,b \in \Bbb R$, $a<b$, and let $f$ be a differentiable real-valued function on an open subset of $\Bbb R$ that contains $[a,b]$. Show that if $\gamma$ is any real number between ...
0
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0answers
22 views

Pointwise and uniform convergence of increasing functions

Let $a< b$ and assume $f_n : [a,b] \to \Bbb R$ are increasing functions, $ n = 1,2,\dots.$ Prove that if $f_n \to f$ pointwise on $[a,b]$, then (i) $f$ is increasing, and (ii) if $f$ is continuous ...
0
votes
0answers
20 views

Lebesgue Measure in relation to product measure

Let $X=Y=[0,1]$ and let $\mathscr M$ be the Lebesgue $\sigma$-algebra on $[0,1]$. Show that any open subset of $X\times Y$ is $\mathscr M\times \mathscr M$ measurable. My approach: By the compactness ...
2
votes
1answer
30 views

a question about differential geometry(Gauss-bonnet theorem and isolated singular point in the surface)

Let C be a regular closed simple curve on a sphere $S^2$. Let v be a differentiable vector field on $S^2$ such that the trajectories of v are never tangent to C. prove that each of the two regions ...