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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0answers
6 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
0
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0answers
9 views

convexity of a function of 2 variables

$f\colon\mathbb{R}\to\mathbb{R}$ is continuous and $|f|$ is convex. Prove that $F\colon\mathbb{R}^2\to\mathbb{R}$ defined as $F(x,y)=|f(x)|+|y-f(x)|$ is convex.
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0answers
7 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
1
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1answer
20 views

How to find the limit points of the set $\{\ a+\alpha\ b \ \ | a,b \in \mathbb Z, \ \alpha \in \mathbb R -\mathbb Q\}$

How to find the limit points of the set $\{\ a+\alpha\ b \ \ |\ a,b \in \mathbb Z \ \text{and}\ \alpha \in \mathbb R -\mathbb Q\}.$ limit point: A point $x$ is said to be a limit point of a non ...
1
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0answers
17 views

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ a real number.Then which is /are true?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ a real number.Then which is /are true? If $n$ is even and $K>0$ , then there exists $x_0 \in ...
1
vote
1answer
11 views

eigenvector of a multiplication operator

Let $\phi\in L^\infty(\mu)$. Define $M_\phi:L^2(\mu)\to L^2(\mu)$ by $M_\phi(f)=\phi f$. What conditions are needed on $\phi$ so that $M_\phi$ has an eigenvector? If $\phi$ is a constant clearly ...
7
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1answer
75 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
2
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2answers
40 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
5
votes
5answers
47 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
0
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0answers
15 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
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0answers
8 views

How to modify Tikhonov regularization?

Consider a linear map $f: X \rightarrow Y$ and let $F$ is a matrix of $f$ and $b$ is one element of $Y$. Our goal is to obtain the element of $X$ corresponding to $b$. In ideal case we can get the ...
2
votes
1answer
38 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
0
votes
0answers
30 views

Proving that region bounded by y=0 and continuous function is a Jordan region.

How do I prove the following? Let $f(x)$ be continuous on $[a,b]$ and let $A=\{(x,y): x \in [a,b] \text{ and } 0 \le y \le f(x)\}$. Prove that $A$ is a Jordan region. I know that I can show ...
0
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1answer
22 views

Uniform Convergence of $(\frac{x}{1+x})^n$

I have an exercise based on the exercise 8 of page 41 in Complex Analysis of Ahlfors. In that exercise they ask for the values of x in which the following series converges: $$ ...
1
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2answers
57 views

Find all continuous real valued function such that $(f(x))^2+C=\int\limits_0^xf(t)dt$

Find all continuous real valued function such that $$(f(x))^2+C=\int\limits_0^xf(t)dt$$ for some $C\in\mathbb{R}$ If I set $F(x)=\int\limits_0^xf(t)dt$ then $F$ is differentiable and ...
0
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0answers
11 views

Upper contingent derivative of a lsc function

Let $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a set-valued map, locally bounded, upper semi-continuous, and taking nonempty, convex and compact values. Let $\,f : \mathbb{R}^n \to ...
2
votes
1answer
26 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
2
votes
2answers
41 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
2
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0answers
48 views

Finding the maximum of two functions with complicated formulas

Let $$ f(\omega)=1+\frac{m(a+\omega^2)}{a^2+\omega^2}+\alpha\left(\frac{a^2+\omega^2-ma}{a^2+\omega^2}\right)\cos(\omega\tau)+\frac{\alpha m\omega}{a^2+\omega^2}\sin(\omega\tau)\;, $$ and $$ ...
1
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0answers
18 views

finding Interior Points of a set

In the normed space $(\mathbb{R}^2, ||(x_1,x_2)||:=|x_1|+|x_2|)$ I want to find Interior Points of $$ \{ (x,1/n) ~~\big|~~ x\in \mathbb{R} \text{ and } n\in \mathbb{N} \}. $$ I guess that the ...
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0answers
29 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
2
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3answers
50 views

Polynomial of degree $2$ has at most $2$ roots

Suppose that $P: \mathbb{R} \rightarrow \mathbb{R}$ is a real polynomial of degree exactly $2$. Prove $P$ has at most two roots. Let $P(x)=a_2 x^2 +a_1 x +a_0$ for all $x \in \mathbb{R}$. I tried to ...
0
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1answer
12 views

How I find a suitable increment value

If I have three variables $x,y$ and $z$, where $x\lt z$ and $y\lt z$, then I need to make each value of $x$ and $y$ equal to or approximately equal to $z$ by adding a ratio of another variable, for ...
3
votes
2answers
80 views

Does $\frac{nx}{1+n \sin(x)}$ converge uniformly on $[a,\pi/2]$ for all $a \in (0,\pi/2]$?

Edit: the question had some missing details. It should read as follows: Prove for all $a \in (0,\frac{\pi}{2}]$, $f_n \rightarrow f$ uniformly on $[a,\frac{\pi}{2}]$. Here $$f_n(x) = \frac{n x}{1 ...
0
votes
1answer
31 views

Bounded sequence of positive numbers

Suppose that $\{x_n\}$ is a sequence of positive real numbers so that $\lim \frac{x_{n+1}}{x_n}=L<1$. Then show that for $n$ large enough and for some $C>0$ we have $0<x_n<Cr^n$. I have ...
0
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2answers
38 views

How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt $

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
0
votes
1answer
36 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
0
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0answers
26 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
2
votes
2answers
55 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
0
votes
1answer
35 views

Lef $f'$ be integrable and $f(0)=0$. Show that $|f(x)|\leq\sqrt{\int\limits_0^1|f'(t)|^2dt}$

Lef $f$ be a fuction such that $f'$ is integrable in $[0,1]$, $f(0)=0$. Show that $$|f(x)|\leq\sqrt{\int\limits_0^1|f'(t)|^2dt}$$ forall $x\in[0,1]$ I did $f(x)^2\leq\int\limits_0^1(f'(t))^2dt$ ...
-2
votes
2answers
20 views

Differential equation maximal interval and solution [on hold]

Consider the differential equation $y' = 1 - y^2$. First, is $y(x) = 1$ the only constant solution? I now want to solve the equation for the initial value problem $y(0) = y_0$, with $y_0 > 1$. ...
4
votes
1answer
36 views

Inverse Function Theorem for Banach Spaces

In the middle of a proof of the Inverse Function Theorem (namely, the proof of Baby Rudin), we use the fact that if $A$ is invertible and: $$ ||B-A||~||A^{-1}|| <1$$ then $B$ is invertible. The ...
2
votes
4answers
57 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
1
vote
1answer
27 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
2
votes
1answer
50 views

Let $f,g$ be differentiable functions such that $\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$. Show that $g(0)=0$

Let $f,g$ be differentiable functions such that $$\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$$ Show that $g(0)=0$ I know it is done if $f(x)=0$ for some $x$, but I just have ...
0
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0answers
27 views

What is the norm for the product of normed spaces?

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite. Is there any natural norm for the ...
5
votes
0answers
33 views

Importance of compactness in Rudin problem.

Okay the problem goes like this: Suppose X, Y, Z are metric spaces, and Y is compact. Let $f$ map X into Y, let g be a continuous one-to-one mapping of Y into Z, and put $h(x)=g(f(x))$ for $x\in ...
1
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1answer
16 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
-1
votes
1answer
38 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
1
vote
2answers
26 views

Show every chain has an upperbound?

Sometimes I feel like proofs like this are pointless. I mean, if we have a partially ordered subset, it seems automatically true that you have a max element. 1) Either you have an infinite sequence ...
2
votes
1answer
37 views

Example for the benefit from monotone convergence

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann ...
0
votes
3answers
59 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
1
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1answer
20 views

does constant convexity assures global minimum

I have the following question: Consider a function $f:R^n \longrightarrow R$, s.t.: there is a point $x_0 \in R^n$ s.t. $\frac{\partial f}{\partial x^k} =0$ $\forall k$. the hessian matrix ...
5
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0answers
21 views

Diffeomorphism-invariant spaces of smooth functions

Let's start with an interesting story. In his celebrated Partial Differential Relations (p. 146), the great Misha Gromov gives a nice exercise of which the following is a (strict) part. Exercise. ...
1
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1answer
40 views

Find a cut-off function in a ball.

Let $0\le r< R\le 1$. How do we find a function $\eta\in C^1(\mathbb{R})$ such that $\eta=1$ in $B_r$ (the ball center at $0$ and radius $r$) and $\eta=0$ outside $B_R$ and $|D\eta|\le ...
2
votes
3answers
21 views

Boundedness theorem question proof check

Here is an attempt at a solution: Since $f(x)>0$, $f(x)>\delta$ for all x between $1$ and $2$ Is this correct?
1
vote
2answers
49 views

Show that $A$ is open in $\mathbb R$

I got this question in a test earlier today. I know it is a very small question, since it only counted 2 marks, but for some reason I simply could not get it?? Let $f:\mathbb R \to \mathbb R$ be ...
1
vote
3answers
47 views

How many continuous involutions on $\mathbb R$ are there? [duplicate]

An involution is a function that satisfies the following: $f = f^{-1}$ MY question is how many involutions can you find in the set of real functions, and how would you go about solving that problem? ...
2
votes
0answers
33 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
-3
votes
1answer
15 views

how to construct a monotonic function on a closed interval which is discontinuous at each end points [on hold]

How to construct a monotonic function on a [0,1] which is discontinuous at each end points?