Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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4
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3answers
138 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
1
vote
1answer
100 views

Prove that $f(x)=x^\alpha$ for all $x>0$.

Suppose that $f:(0,\infty) \to \Bbb R$ is differentiable and that there exists a constant $\alpha$ belonging to $\Bbb R$ such that $x*f'(x)=\alpha*f(x)$ for all $x>0$ and $f(1)=1$. Prove that ...
3
votes
2answers
93 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
1
vote
2answers
89 views

Prove that $f(x)=x$ for all $x\geq 0$

Suppose that $f$ is continuous on $[0,\infty)$ such that $f(x)>0$ for all $x>0$ and that $f^2(x)=2\int_0^x f(t)\,dt$ for all $x>0$. Prove that $f(x)=x$ for all $x\geq 0$. Attempt at a ...
3
votes
1answer
53 views

Homotopy vs Conservative

I learned about conservative fields in multivariable calculus. I'm always curious about finding other or more fundamental methods of describing a concept (or concepts) in math, and better ...
1
vote
1answer
34 views

function of class $C^r $

Let $f,g: \mathbb{R}^2 \to \mathbb{R}$ such that $g(x,y)=f(x,y)+(f(x,y))^5$. If $f$ is continuous and $g$ is class $C^r$, show that $f$ is a function of class $C^r$ and calculate $df/dx$.
0
votes
3answers
58 views

Continuity and sequences problem

Question: Suppose $ f: \mathbb{R} \rightarrow \mathbb{R} $ is a function satisfying: $\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty $ part a) Show that ...
1
vote
0answers
85 views

Mean value theorem

I have a small question: We consider the existence of multiple solutions of the periodic boundary value problem $$\ddot{u}(t)+\nabla F(t,u(t))=0\\ u(0)-u(T)=\dot{u}(0)-\dot u(T)=0\tag{8}$$ where ...
0
votes
1answer
119 views

Proving $\mathbb R^2 \setminus \mathbb Q^2$ is connected [duplicate]

I am trying to prove that the set $\mathbb R^2 \setminus \mathbb Q^2$ is connected. I don't know if the following is true: could it be that it is also path connected? If that is the case, maybe it's ...
0
votes
1answer
61 views

$f: \mathbb{R}^n \mapsto \mathbb{R} $ is strictly increasing

While I'm well aware of what "strictly increasing" means for a function that maps from $\mathbb{R}$ to $\mathbb{R}$, I'm unclear about what is meant by $f: \mathbb{R}^n \mapsto \mathbb{R} $ is ...
0
votes
2answers
705 views

What is an example of a lower semicontinuous function that is not continuous?

I couldn't find an example anywhere. Does anyone know such example? Thanks.
2
votes
0answers
90 views

Limit of $\ln(1\cdot\ln(2\cdot\ln(3\cdot\ln(4\cdots))))$

I recently asked for the limit $\lim_{n->\infty} \ln(1+\ln(2+\ln(3+\ln(4+\cdots+\ln(n))\ldots)$. But what about the similar limit $\lim_{n->\infty} \ln(1\cdot \ln(2\cdot \ln(3\cdot ...
2
votes
5answers
224 views

Showing a set is closed

I have to show that the set $ A \subset C[0,1]$ defined by $A = [f \colon 0 \leq f(x) \leq 1 \forall x \in [0,1] ]$ is closed in the $||.||_\infty$ norm. Now i know the definition for open/closed set ...
0
votes
1answer
52 views

Contraction of a differential equation and its mapping

Consider a differential equation $\frac{dx}{dt}(t) = 2cos(tx^2(t))$ with initial condition x(0) =1. Check that the solutions can be found as fixed points of the map such that $f(t) : t \in [0,T]) ...
2
votes
1answer
124 views

Baby rudin chapter 6 exercise 14 ---Isn't it a typo?

$$ f(x) \ = \ \int_{x}^{x+1} \sin(e^t) \, dt. $$ Show that $$ e^x | f(x) | \ < \ 2 $$ and that $$ e^x f(x) = \cos(e^x) - e^{-1} \cos(e^{x+1}) + r(x), $$ where $|r(x)|< C e^{-x}$ for some ...
0
votes
1answer
147 views

question involving double sequence of real numbers

I'm trying to do the following problem. I'm working on section 2.5 (product measures) of Folland's Real Analysis book. This problem doesn't come from the book, though. For this problem, I'm ...
1
vote
1answer
51 views

Composition of continuous linear maps is also a continuous linear map

Let $V, W , X$ be normed spaces and let $T \colon V \to W$ and $ S \colon W \to X$ be continuous linear maps. show that $ S \circ T \colon V \to X$ is a continuous linear map and that $||S \circ T || ...
1
vote
2answers
69 views

How to calculate the Maclaurin series for $\frac{{x - \sin (x)}}{{{x^2}}}$

For the function $$f(x)=\frac{{x - \sin (x)}}{{{x^2}}},\quad x\neq 0$$ it is known that the Maclaurin series of the function $f(x)$ of class $C^\infty$ is equal to the corresponding Taylor series ...
2
votes
2answers
70 views

Bounded sets in $\Bbb{R}^n$

Let $S\subset \Bbb{R}^n$ is a bounded set. I want to show that $\underline{\underline{the}}$ sphere(closed ball) of $\textbf{smallest radius}$ which contains S is $\underline{unique}$. the assumption ...
1
vote
1answer
51 views

Normed vector spaces and operator norm

Let $T \colon V \to W$ be a linear map between normed vector spaces $(V, \parallel \cdot \parallel_V)$ and $(W,\parallel \cdot \parallel_W)$ as $$\parallel T \parallel :={\rm sup}\ \{ \parallel T(x) ...
2
votes
1answer
61 views

The contraction mapping theorem

Let $f : \mathbb{R} \to \mathbb{R} : f(x) = 1 + x + \mathbb{e}^{-x}$ and let $ U [1,\infty)$. Firstly i need to show that $f$ maps $U$ into itself. But im can only see how $f$ maps $U$ to $[2 ...
1
vote
2answers
152 views

Why does the set of rational numbers be a countable union of closed sets?

Im reading Chapter9 Category, Real Analysis, Carothers, 1ed, talking about discontinuous functions of metric space,: At the bottom, the author claimed that Q is $F_σ$ set, which means that the set ...
1
vote
3answers
65 views

Do closed intervals exist?

If 0.999.. = 1, does that mean that infinitesimals are not allowed in $(-\infty,1)$? Otherwise, we would have $0.9 \in (-\infty,1)$, $0.99 \in (-\infty,1)$, $0.999\in(-\infty,1)$, ad infinitum.
1
vote
1answer
78 views

Introduction to Analysis: L'Hospital's Rule

For class, we are to prove L'Hospital's Rule for $\infty/\infty$ case, Let $\lim_{x\rightarrow\infty}\frac{f'(x)}{g'(x)}$; choose a so that $\frac{f'(x)}{g'(x)}\approx_{\epsilon}L$ for $x>a$. ...
1
vote
1answer
131 views

How prove this only point such $f(x,y)$ obtain the maximum

Question: let $$D=\{u=(x,y)\in R^2\colon||u||=\sqrt{x^2+y^2}\le\dfrac{1}{2}\}$$ and $f(u)=f(x,y)$ is all plane continuously differentiable,and such $$||\nabla f(0,0)||=1,||\nabla f(u)-\nabla ...
0
votes
1answer
34 views

Different compositions with nearly the same values

Consider the following compositions $$\cos(\cos(\cos(\cos(\sin(1))))) = 0.7605544662971730378084837618$$ $$\cos(\sin(\cos(\cos(\sin(1))))) = 0.7599347639070954684181715364$$ The length of such a ...
0
votes
1answer
231 views

Relation of total variation of a function $f$ and the integral of $|f'|$

Let $f:[a,b] \to \mathbb R$ a function of class $C^1$ on the interval $[a,b]$. Prove that: i) $f$ is a function of bounded variation. ii) The equality $V_a^b f= \int_a^b |f'(x)|dx$ holds. My ...
1
vote
2answers
62 views

Exchange the order of limit

This is a problem I met when doing the homework, consider a positive sequence $\{a_{ij}\}$, if for each $i$, $\sum\limits_{j=1}^{\infty}a_{ij}$ exists and $\leq1$, for each $j$, ...
2
votes
1answer
57 views

Defintion of $\ell^\infty$

I have come across the space of bounded sequences denoted as $\ell^\infty$ in my course, but not a clear, concise definition. I have seen sometimes when these includes sequences in $\mathbb{R}$ that ...
1
vote
1answer
45 views

Demonstrate that the following metric space is not compact

Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact. I know that sequentially compact and ...
1
vote
1answer
61 views

IFF conditions for convergence in probability and almost surely

I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ...
2
votes
1answer
277 views

The set of discontinuous points is countable union of closed sets

Im reading chapter9 Category, Real Analysis, Carothers, 1ed, talking about discontinuous functions of metric space. Here is a proof for a theorem that the set of discontinuous points is the countable ...
1
vote
1answer
128 views

Law of large numbers for Brownian Motion

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? ...
1
vote
2answers
281 views

For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
2
votes
1answer
44 views

Converge of distance between two sequences

Let $(s_n)$ and $(t_n)$ be sequences in the metric space $(X,d).$ Suppose $s_n \rightarrow s \in X$ and $t_n \rightarrow t \in X.$ Prove that $d(s_n,t_n) \rightarrow d(s,t).$ My attempt: Assume $s_n ...
4
votes
1answer
125 views

There exist $\{a_{n}\},\{b_{n}\}$ such $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=c?$

Let $ A$ and $ B$ are two infinite subsets of the natural numbers $\mathbb{N}$, such that $$A\cap B=\emptyset \qquad A\cup B=\mathbb N$$ Question: is it true that for every natural $c>0$, ...
2
votes
2answers
56 views

Does “$\exists \delta >0$ S.T $||x(0)-x_e||<\delta\Rightarrow \displaystyle \lim_{t\rightarrow \infty}||x(t)-x_e||=0$” imply stability?

Recall the definition of stable and Asymptotically stable: A fixed point $x_e$ of a vector field is called (Lyapunov) stable if $\forall \varepsilon>0,\exists \delta(\varepsilon)$ such that ...
1
vote
1answer
107 views

Riesz representation theorem application

I need help with the second question below My thoughts: Taking the first functional, by the riesz representation theorem, I can find a signed measure $v$ on $B[0,1]$ such that $l_1(p)=$ the ...
2
votes
1answer
40 views

Proving Riemann Integrability- Uniform Convergence

Ok so I have come across a proof to show if we have a sequence of functions $f_n$ converging uniformly to $f$ say in the reals, such that if $f_n$ is riemann integrable then so is $f$. In the proof ...
0
votes
1answer
81 views

Definition of divergence (negation rules)

Some background before my question. A question in my homework is as follows: Using no negative words, say what it would mean for a sequence $\langle a_n \rangle$ to diverge. Our definition of ...
2
votes
1answer
74 views

Find the minimum value of the expression.

Find the minimum value of the expression. $x,y,z \in R$ $\sqrt{x^2+1}+ \sqrt {4+(y-z)^2} + \sqrt{1+ (z-x)^2} + \sqrt{9+(10-y)^2}$
9
votes
2answers
164 views

Maximum of $\frac{\sin z}{z}$ in the closed unit disc.

I have some trouble with the following question: Let $$f(z)=\frac{\sin z}{z},\quad\text{for }z\in\mathbb{C}.$$ What is the maximum of $f$ in the closed unit disc ...
3
votes
4answers
72 views

Evaluating $\lim_{n\rightarrow \infty} \frac{n\sqrt{\ln n}}{(n+1)\sqrt{\ln(n+1)}}$

I'm supposed to look at $$\lim_{n\rightarrow \infty} \frac{n\sqrt{\ln n}}{(n+1)\sqrt{\ln(n+1)}}$$ which is of course the result of a ratio test. While Wolfram Alpha tells me the limit is $1$, I don't ...
0
votes
2answers
36 views

How do I show that this expression is greater than $0$?

I'm trying to show an expression is greater than $0$. The expression is $\frac{1}{n+1}+\log(\frac{n+1}{n+2})$. I can't really get anywhere with this, and any help would be much appreciated. Thanks.
0
votes
1answer
51 views

Why is this transformation true?

I have just a simple question i think. I tried to implement the $\chi^2$-test.I have this document where it is said on page 41, that I have to implement the test like this:$$\frac{\sum_{0\leq i < ...
0
votes
2answers
448 views

Ratio test - Ratio tends to infinity

I have to prove that if $a_n\neq0$ for all n, and $|\frac{a_{n+1}}{a_n}|\to \infty$, then $\sum a_n$ diverges. I have proved the ratio test for when the limit of the ratio, L, is less than 1 or ...
21
votes
2answers
870 views

How to prove there exists $c$ such $f(c)f'(c)+f''(c)=0$

Nice Question: let $f(x)$ have two derivative on $[0,1]$,and such $$f(0)=2,f'(0)=-2,f(1)=1$$ show that: there exist $c\in(0,1)$,such $$f(c)f'(c)+f''(c)=0$$ my try: since ...
2
votes
1answer
354 views

Let $f,g$ be differentiable with $f(0)=g(0)$ and $f'(x)<g'(x)$. Prove that $f(x)<g(x)$.

Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $f(0)=g(0)$ and $f'(x) < g'(x)$ for all $x$ belonging to the set of real numbers. Prove that $f(x)<g(x)$ for all $x>0$. Any ...
1
vote
1answer
71 views

A surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$

Suppose that a surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$. Let $$\tilde E d\tilde u^2+ 2\tilde F d\tilde ud\tilde v+\tilde G d\tilde ...
1
vote
0answers
47 views

Maximal value of an infinite set. [duplicate]

Consider a continuous function $f$ over an interval $[a,b]$. Let $S$ be the set of all values that $f(x)$ takes over $I$. Intuitively speaking, I believe this set has a maximal and minimal value. Is ...