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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0
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1answer
19 views

Riemann integral property proof using the definition

We say that a function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for every $\epsilon>0$, there are two step functions $g_1,g_2$ such that $g_1 \leq f \leq g_2$ and $\int_a^b ...
0
votes
2answers
92 views

I can use MVT on $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

If I can use MVT: $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt=x\cdot f\left(c\right)$ when $x\rightarrow \infty ,\:c\rightarrow \infty $ so we'll have to evaluate $\lim _{x\to \infty }x\cdot ...
0
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0answers
20 views

If $d(x_0,y_j)\to d(x_0,y_0)$, then $y_j \to y_0$.

Consider a metric space $X$, and a compact subset $C\subset X$.Let $x_0\in X-C$. We can show that there is a point $y_0\in C$ such that $d(x_0,y)=\inf_{y\in C} d(x_0,y)$. Now suppose there is ...
2
votes
1answer
18 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
0
votes
1answer
34 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
4
votes
1answer
43 views

If the derivative is $0$, then $f$ is constant in a banach space

My question is simple. Take a differentiable function $f: U \subset \mathbb{E} \rightarrow \mathbb{F}$, where $\mathbb{E}, \mathbb{F}$ are banach spaces and $U$ is an open connected subset of ...
0
votes
1answer
14 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
3
votes
4answers
105 views

Find the limit $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$

Find the limit if it exists $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$$
0
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0answers
11 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives { f'(x)$\in L(R^{^{n}},R^{^{m}}):x\in$ [a,b] }is ...
2
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0answers
33 views

(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
0
votes
1answer
13 views

Is this function differentiable w.r.t. a variable in an indicator?

I have $$y^i = x^i - \alpha \sum_{j \epsilon N} (x^j - x^i) I_{x^i \lt x^j} - \beta \sum_{j \epsilon N} (x^i - x^j) I_{x^i \ge x^j}$$ where N = {1, 2, ..., n}, and $I_{x^i \lt x^j}$ is 1 when $x^i ...
0
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2answers
315 views

Dedekind Cut additive inverse

Let $\alpha$ be Dedekind cut and define $\alpha^* :=\{x\in\mathbb{Q}|\exists r>0\space \text{such that} -x-r\notin\alpha\}$. I need to show that $\alpha^*$ is a Dedkind Cut and the additive inverse ...
1
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0answers
62 views

On definition of Riemann integral

Let $I = [a, b]$ be the finite closed interval of $\mathbb R$. A partion $P$ of $I$ is a finite sequence $a = a_0 \lt a_1 \lt ... \lt a_n = b$. We write $P\le Q$ if $P \subset Q$ where $P, Q$ are ...
1
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1answer
29 views

Integral relations when using different measures.

Let $(X,\mathcal{M})$ a measurable space and $\mu$,$\nu$ two non-negative measures s.t $\mu \geq \nu$. Does it hold that $\int_E f \, d\mu \geq \int_E f \,d\nu $ where $E \in \mathcal{M}$. I suspect ...
2
votes
1answer
41 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
5
votes
1answer
30 views

If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
0
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1answer
31 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 ...
0
votes
2answers
43 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 ...
2
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1answer
19 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 ...
-2
votes
2answers
50 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 ...
2
votes
1answer
35 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 : ...
-1
votes
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 ...
0
votes
1answer
22 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
0
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0answers
27 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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0answers
13 views

Kernel function related proof [on hold]

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 = ...
2
votes
2answers
75 views
+50

What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
1
vote
2answers
33 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 ...
3
votes
2answers
107 views

Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$

Suppose $f,f_n$ are measurable and uniformly bounded on $[a,b]$. Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$ Attempt: We note that since $f$ and $f_n$ are bounded and are ...
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.. ...
-1
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1answer
26 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 ...
0
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0answers
17 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.
1
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3answers
28 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
votes
1answer
21 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: ...
-3
votes
3answers
32 views

A problem of Schwarz derivative [on hold]

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 ...
4
votes
1answer
90 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 ...
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 ...
4
votes
1answer
31 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 $a \in E$ ...
1
vote
1answer
76 views

If a map on a complete metric space has a contraction property, it has a unique fixed point

I am stuck on the following problem: Prove that if $(X, d)$ is a complete metric space and $f : X\rightarrow X$ is a function with the property that there is a number $A < 1$ such that ...
7
votes
3answers
236 views

What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$?

I want to determine the limit given below: $$\lim_{n\to\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}}$$ I have tried to solve thise several times ,but with no results.I have tried using lema stolz ...
1
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0answers
26 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 ...
1
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1answer
19 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 ...
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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
2
votes
0answers
27 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 ...
0
votes
1answer
18 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?
6
votes
1answer
202 views

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. Let $f:X\to Y$ and $a\in X$ There is a map $\Delta : X \to L(X,Y)$ continuous at ...
0
votes
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
votes
0answers
22 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: ...
0
votes
4answers
46 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 ...
1
vote
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 ...
0
votes
2answers
21 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 ...