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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4answers
204 views

About $ \lim_{x\rightarrow 0}\frac {\sin x}{x} = 1$ [duplicate]

I do not understand how $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ As if $$ x = 0, \frac{\sin (0)}{0} = \frac {0}{0} $$ So if someone could explain this I would appreciate it! Thanks!
1
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3answers
174 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
1
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1answer
129 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
1
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0answers
935 views

The Cantor set is nowhere dense

I am considering the so called Cantor ternary set $C$ on $[0,1]$. I have just proved that its Lebesgue measure is $0$. To show that $C$ is nowhere dense, is it correct the following reasoning? By ...
6
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2answers
131 views

If $f^3=\rm id$ then it is identity function [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f^3(x)=f\circ f\circ f(x)=x$ for all $x$. How can I prove $f$ is the identity function?
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1answer
109 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
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3answers
166 views

Non-decreasing functions and continuity

I have the following situation: $f\colon\mathbb{R}\to\mathbb{R}$ is a non-decreasing $g\colon\mathbb{R}\to\mathbb{R}$ is defined as $\ g(x):=\lim_{t\to x^+}f(t)$ I have proved that also $g$ is ...
1
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1answer
31 views

Connected Subset of Finite Topological Space

What I want to know is this: Am I correct in thinking that if we have a finite topological space (so no reals or anything here), that any connected subset contained in this space has only one ...
2
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4answers
232 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
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1answer
75 views

Integrability of sums of Dirac deltas

this is my first post in the forum and I am an engineer, so I apologize in advance if my question is not clearly stated. Consider the function $f(x)=\sum_{i=1}^N a_i\delta(x-x_i)$ where ...
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1answer
33 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
3
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0answers
62 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
0
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3answers
55 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
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0answers
54 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
0
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1answer
465 views

slope of curve represented by discrete points

I have data which are visualized in this chart: I need to compute slope of increasing / decreasing parts of the curve. I can't use any 2 points because of noise in data. Maybe numerical derivative ...
7
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2answers
320 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
5
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2answers
176 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
1
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1answer
58 views

Differentiability of a convex function

Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be convex functions such that $f\ge g$ and $f(0)=g(0)$. Show that if $f$ is differentiable in 0, then $g$ is too and $$ f'(0)=g'(0)$$ I have no idea ...
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0answers
60 views

Path test versus iterated limits for proving existence of a limit

Which is the strongest method for proving existence of limits (or the non-existence of a limit), the path test or using iterated limits? Indeed, there are cases where the iterated limit shows that ...
6
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2answers
123 views

Is $f$ surjective, where $|f(x)-x| \leq 2$?

Let $f$ be continuous and $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $|f(x)-x| \leq 2$ holds for all $x$. Is $f$ surjective?
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1answer
17 views

Finite convergence radius

can somebody explain the fact that an analytic function like arctan(x) ( it can thus be written as a power series ) on the whole R, has a finite convergence radius ( and not an infinite one as one ...
0
votes
1answer
22 views

Decay of fourier transform of a function with two variables

In the paragraph about 'use in harmonic analysis', the modulus of fourier transform of a (one variable) function $f(\xi)$ is shown to be bounded by a quantity involving powers of $\xi$. ...
5
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1answer
162 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
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1answer
181 views

consequences of Schwarz lemma of holomorphic functions of unit disk

Let $D$ be the open unit disk centered at $0$ in the complex plane. Let $f:D\longrightarrow D$ be holomorphic such that $f(0)=0$. Use the Schwarz lemma to prove that $|f(z)+f(-z)|\leq 2|z|^2$ for any ...
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2answers
746 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
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2answers
63 views

Fejer Kernel is Unbounded

Statement: Given the Fejer Kernel $F_n(x) = \frac{1}{n}\bigg(\frac{\sin(\frac{nx}{2})}{\sin(\frac{x}{2})}\bigg)^2$. Show that $F_n(x)$ is unbounded for $x=0$ as $n\rightarrow \infty$
0
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1answer
61 views

Some properties of capacity

Let $\Omega\subset\mathbb{R}^N$. For $K\subset \Omega$ we can define the $p$-capacity, $p\in (1,\infty)$ as the number $$\operatorname{cap}_p(K)=\inf \int_\Omega |\nabla u|^p$$ where the infimum is ...
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2answers
79 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
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2answers
74 views

An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$. I am trying to find a $T$ such that $\det(J_T)=1$. (*) The trivial cases are $T(x,y)=(x,y)$, ...
3
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0answers
65 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + ...
2
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2answers
54 views

Verifying if a function is Lipschitz

Let $\Omega\subset \mathbb{R}^N$ be a domain with $N\ge 2$. Let $K\subset \Omega$ be a compact set and take $u:\overline{\Omega}\to\mathbb{R}$ such that $u$ is Lipschitz and $u=1$ in $\partial K$. ...
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0answers
68 views

Analysis over finite fields

At the beginning of my study in analysis I learned something about convergence of sequences for example, metric spaces and so forth... Most of the time we considered metric spaces $(\mathbb{K}, d), ...
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1answer
57 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
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3answers
416 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
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1answer
55 views

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$?

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$? That is, is the integral $$\int_{S^{n-1}}\frac{1}{\sqrt{|x_1|}}d\sigma(x)$$ finite? Where $\sigma$ ...
0
votes
1answer
40 views

Bump function's support

How does this function (for instance) have compact support, if the support is the open interval (-1,1) ? http://en.wikipedia.org/wiki/Bump_function
13
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2answers
450 views

Finite dimensional subspace of $C([0,1])$

Let linear $S$ be a subspace of $C([0,1])$, i.e., the continuous real-valued functions on $[0,1]$. Assume that there exists $c>0$, such that $\|\,f\|_\infty\leq c \|\,f\|_2$, for all $f\in ...
2
votes
1answer
83 views

Partitions of Unity-Integration on Manifolds

So lets say I have a $k$-manifold $M$ in $\mathbb{R}^n$, and I cover it up with coordinate patches $\{\alpha_i\}$. I can find a set of partitions of unity $\{\phi_1,...\phi_l\}$ on $M$ which is ...
0
votes
1answer
47 views

an analysis problem continuous function

Let $f\in C^{\infty}(R,R)$ infinitely differentiable on real line, $f(x)=1$ for $ x\in[-1,1]$, and $f(x)=0$ for all $x\notin(-2,2)$. Prove that for any $C>0$, there exist an integer $n\ge0$, and a ...
1
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0answers
34 views

Diffeomorphism/Problem/Euclidean spaces

Problem: Let $f$ : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ be a $C^{1}$ function such that |$f'(t)$| $\leq$ $k$ < $1$, $\forall$ $t$ $\in$ $\mathbb{R}$. Define $\phi$ : $\mathbb{R}^{2}$ ...
2
votes
1answer
321 views

Example of a function continuously differentiable that is bijection from R to R, but the continuous inverse is not differentiable

Does there exist a function f: R-> R, f is continuously differentiable and bijection and has a inverse function g: R-> R, but g is not differentiable everywhere? I compare it with the inverse ...
0
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1answer
468 views

Riemann-integrable functions and pointwise convergence

Hello, I was hoping for some advice on finding a function which will satisfy this. I think I am okay with the actual execution of the answer, but I don't know how I'm supposed to find a suitable ...
0
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1answer
40 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
0
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1answer
29 views

conditioon that is necessary and sufficient to ensure $(a_n)_{n=1}^{\infty}$ of integers converges via $\epsilon - N$ logic

i want to extend the knowledge the previous question of mine obtained and know when a sequence of integrs does actually converge. i want to work this out using $\epsilon - N$ logic. my thoughts: this ...
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2answers
854 views

$\epsilon - N$ definition of a limit of sequence problem

i have a question i cannot seem to solve! i would really appreciate help if possible. please explain how to solve this question from textbook, i really want to learn but i cant $$\lim \limits_{n \to ...
0
votes
1answer
51 views

$1/z$ continuous on complex plane minus origin

Is $1/z$ continuous on $\mathbb C \setminus 0$? I can prove it's continuous on $\mathbb C \setminus B(0,c)$: $$ \left | {1/z - 1/y } \right | = \left | {y-z \over yz } \right |$$ and if $z,y \ge c ...
1
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1answer
41 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
1
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2answers
143 views

Analysis problem from Romanian Contest - 2 sequences which forms another one

Let $a,b$ be 2 real numbers, and the sequences $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}$ defined by $a_{1}=a$, $b_{1}=b$, $a^2+b^2 <1$ and \begin{cases} ...
0
votes
4answers
144 views

Proving limit using $\epsilon - N$

I am not sure I understand the $N - \epsilon$ method for proving the equality of a limit. I have a past mid-semester exam question that has: $$\lim \limits_{x \to 1} (x^2 - 4x) = -3$$ Now it seems I ...
0
votes
2answers
48 views

Prove that $f^y$ and $f_x$ are Lebesgue-integrable

Let $f:\Bbb R^2\to \Bbb R$ given by: $$f(x,y) = \begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2} & \text{if $(x,y)\in(0,1)\times(0,1)$} \\ 0 & \text{if $(x,y)\not\in(0,1)\times(0,1)$} \\ ...