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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2
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1answer
26 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
3
votes
2answers
15 views

Show that any subset of $(\mathbb{N},d)$ is open and closed

Show that any subset of $(\mathbb{N},d)$ is open and closed, where $$d(m,n) = \frac{|m-n|}{1+|m-n|}$$ my attempt: let $A \subset \mathbb{N}$ then for any $x \in A$ we have that $B(x,1/3) = \{x\} ...
0
votes
0answers
19 views

Integrating 2 form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
1
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0answers
11 views

Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?
1
vote
2answers
13 views

If $x \in (b, \infty)$ Show there exists a natural number $n_0$ such that $x > b + \frac{1}{n_0}$

Assume $\displaystyle \lim_{n \to \infty} \frac{1}{n} =0$ and for $b \in \mathbb{R}$, $\displaystyle \lim_{n \to \infty} b = b$. WITHOUT using the Archimedean postulate, show that if $x \in (b, ...
3
votes
1answer
50 views

$\int_a^b f(x) g'(x) dx = 0$ implies $f$ is constant

Given $f$ is continuous on $[a,b]$, $\forall g$ which is a continuously differentiable function on $[a,b]$, with $g(a)=g(b)=0$, the following equation is satisfied: $\int_a^b f(x) g'(x) dx = 0$. I ...
1
vote
2answers
59 views

Show that the antiderivative exist [on hold]

I am new to this. How do I show that the antiderivative exist and show that is continuous too? Thanks
1
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1answer
53 views

Explanation for absolute value

So $f_a:R\rightarrow \:R,\:f_a(x)=\:\frac{1}{\left|x-a\right|+3}$, and we have to evaluate $\lim _{a\to \infty }\int _0^3\:f_a\left(x\right)dx$. But $\left|x-a\right|\:$ is equal with: ...
0
votes
0answers
15 views

Question on Limits; Invariance Principle

This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. "E1. Starting with a point S (a, b) of the plane with 0 < b < a, we generate a ...
1
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3answers
43 views

Limit function of $\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$

What is the limit of the sequence of functions$\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$? I think this just doesn't exist, since it goes to $\infty$ in $x=0$ and goes to $0$ everywhere ...
0
votes
1answer
25 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
0answers
35 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
20 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) ...
4
votes
0answers
46 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 $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
0
votes
1answer
16 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 ...
2
votes
0answers
36 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 ...
4
votes
2answers
62 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

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 ...
1
vote
1answer
30 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 ...
5
votes
1answer
36 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. ...
2
votes
1answer
21 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
1answer
40 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 : ...
0
votes
2answers
51 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
vote
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 = ...
1
vote
2answers
34 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 ...
0
votes
0answers
35 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 ...
0
votes
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 ...
1
vote
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.. ...
0
votes
0answers
22 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
vote
3answers
30 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
22 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
33 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
33 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
0answers
28 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 ...
-5
votes
0answers
37 views

Which is strongest of all? [on hold]

lipschitz condition, uniformly continuous, differentiable and which is weakest? please help
1
vote
1answer
20 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
votes
0answers
28 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
votes
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 ...
0
votes
0answers
25 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: ...
-1
votes
0answers
38 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
20 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?
-3
votes
0answers
19 views

the Fisher equation has no positive traveling wave solution [on hold]

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 $U(-\infty)=0$
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 ...
0
votes
0answers
15 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
vote
3answers
36 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 \{ ...
0
votes
4answers
47 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
votes
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
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
1answer
26 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 ...
0
votes
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 ...