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).

learn more… | top users | synonyms (1)

5
votes
2answers
127 views

Proving the limit at $\infty$ of the derivative $f'$ is $0$ if it and the limit of the function $f$ exist. [duplicate]

Suppose that $f$ is differentiable for all $x$, and that $\lim_{x\to \infty} f(x)$ exists. Prove that if $\lim_{x\to \infty} f′(x)$ exists, then $\lim_{x\to \infty} f′(x) = 0$, and also, give an ...
0
votes
1answer
37 views

If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?

If $f \in L_2(a,b)$, then I want to show that the antiderivative $$ F(x) := \int_a^x f(y) d y $$ is in $L_2$ (I guess this is true). If $L_2(a,b)$ would be closed under pointwise product, i.e. if $f,...
4
votes
1answer
155 views

Theorem 4.22 from baby Rudin: continuity and connectedness

I have some parts that I don't understand from the given proof. The theorem is: If $f$ is a continuous mapping of a metric space $X$ in to a metric space $Y$, and if $E$ is a connected subset of $X$, ...
1
vote
2answers
64 views

Clarification about notation for one-sided limits

Is $\lim_{x \to 3-0} f(x)$ the same as $\lim_{x \to 3^-} f(x)$, and is $\lim_{x \to 3+0} f(x)$ the same as $\lim_{x \to 3^+} f(x)$? Could anyone clarify this for me please? Thanks
1
vote
2answers
47 views

Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
3
votes
0answers
40 views

Poincare map trouble

Consider $ X' = F(X)$, $F \in C^1(\mathbb{R}^2)$. Suppose that the system has an orbit $\mathcal{O}_p$ and $\Sigma$ an transversal section in $P$. Show that if $$\pi^{n+1}(\Sigma) \subset \pi^{n}(\...
3
votes
2answers
130 views

Arzela-Ascoli Anthony Knapp Proof

STATEMENT: (Arzela Ascoli Theorem) If $\left\{f_n\right\}$ is an equicontinuous family of scalar-valued functions defined on a compact Hausdorff space $X$ and if $\left\{f_n\right\}$ has the property ...
0
votes
0answers
111 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral $$\int_{-\infty}^{...
0
votes
0answers
14 views

Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
1
vote
1answer
372 views

Order of study? Rudin, Spivak, Munkres?

I'm currently taking an analysis course at a top 10 four year university in which we use Baby Rudin as our primary text. I was curious to know the order in which I should continue my studies. That ...
3
votes
2answers
54 views

Is $[0,1]^2 \setminus \{(a,b)\}$ connected?

I am pretty sure that this set is in fact connected but I am struggling to see how to prove it, it is simple to see that $[0,1] \setminus \{x\}$ is disconnected but I can't see how to relate ...
1
vote
2answers
45 views

Let $f(x)$ be continuous on $[0,1]$. Also, let $\int_0^1{f(x)x^n dx}=0$ for all $n\geq 0$. Prove that $\int_0^1{f^2(x)}=0$.

Let $f(x)$ be continuous on $[0,1]$. Also, let $\int_0^1{f(x)x^n dx}=0$ for all $n\geq 0$. Prove that $\int_0^1{f^2(x)}=0$. My hunch is that if $\int_0^1{f(x)x^n dx}=0$ for all $n\geq 0$, then $f(x)=...
1
vote
2answers
77 views

The function $\frac1x$ is an homeomorphism

I have the function $f:(0,+\infty)\rightarrow (0,+\infty)$ defined by $f(x)=\frac1x$ I want to prove that $f$ is an homeomorphism. So I have that $f$ is surjective or onto by definition and that $f$ ...
1
vote
1answer
124 views

change the order of lim and sup

There is a sequence of nonnegative real valued functions $\{f_n(t)\}$ which are bounded $|f_n(t)|\le 1 ,\forall n \forall t\in[0,1]$ and $\forall n$ $$\lim_{t\to 0}f_n(t)=0\tag{1}$$ Could we get $$\...
2
votes
3answers
118 views

Continuous functions and infinum

Let $f:\mathbb R \to \mathbb R$ with $f(-2)=4$ and $f(3)=7$. Let $S:=\{x \in [-2,3]\mid f(x)\geq 5\}$. Then $\alpha:=\inf S$ exists. If $f$ is continuous at $\alpha$, show that: (a) $-2<\alpha<...
4
votes
4answers
94 views

How to prove that $\ln x\leq x-1 \forall x>0$?

I need to prove that $\ln x\leq x-1 \forall x>0$, using the Mean value theorem. For $x=1$, the equation is true. So, for starters I'll check for $x>1$. By applying the aforementioned ...
2
votes
1answer
70 views

Sum in terms of $e^x$

Is it possible to write the sum $$ \sum_{k=0}^{\infty} 2^{2k-1} \frac{x^k}{k!}$$ in terms of $e^x$? That is, using the fact that $$e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$$
0
votes
1answer
68 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
8
votes
3answers
111 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
0
votes
0answers
17 views

Assume that $y\in l^3$. Show $Y_n=(b_{n+k})_{k\geq1}$ converges to $0$?

Assume that $y=(b_1,b_2,b_3,..)\in l^3$. Show $Y_n=(b_{n+1},b_{n+2},b_{n+3},...)=(b_{n+k})_{k\geq1}$ converges to $0$ in $l^3$ with norm $\|\cdot\|_3$ Help! I understand $b_n \rightarrow 0$ and ...
4
votes
1answer
48 views

When is an oscillating integral small?

I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int f(x)\,e^{ikx}\,dx=...
1
vote
1answer
48 views

Applications of Singular Functions

For our purposes here, a singular function is a continuous function such that the part which is absolutely continuous with respect to Lebesgue measure is zero. For example, the Cantor function or "...
2
votes
1answer
43 views

Uniform Convergence of a sequence so that $lim_{n\to\infty} { A_{n+1}\over A_n} = L$

Suppose that {${A_n}$}$_{n=1}^\infty$ is a sequence of numbers so that $lim_{n\to\infty} { A_{n+1}\over A_n} = L$ and $0 < L < 1$. Show that the sequence of functions $g_n(x) = \sum_{k=1}^n ...
5
votes
2answers
72 views

deriving the sum of $x^n/(n+2)^2$

I am writing a research paper and I have stumbled upon an issue. I have to evaluate $$\sum_{n=1}^{\infty} \frac{x^n}{(n+2)^2}$$ Here is what I did: $$ \sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-...
0
votes
0answers
26 views

Re-parametrization of triangle

Let $\beta(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a closed parametric curve. For example, let $\beta(t)$ is a circle. So one parametrisation may be $\beta(t)=(x(t),y(t))'$ where $x(t)=2\pi \...
0
votes
1answer
26 views

If f and g are inverses of each other, using the chain rule show $f′(x)= {1 \over g′(f(x))}$.

Use the chain rule to show that if f and g are inverses of each other then: $f′(x)= {1 \over g′(f(x))}$. I know that I have to use $ g(f(x)) = x$, but I am not really sure where to start.
0
votes
1answer
50 views

Need some help with complex-analysis definitions and understanding

Right, so I'm struggling proving/disproving that for functions $u,v: \mathbb R^2 \to \mathbb R$ if $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (so the relation is ...
30
votes
12answers
1k views

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
1
vote
0answers
38 views

Inequality of finite sequences of real numbers .

Is the following inequality true for real numbers $\lambda_{i}$ and $\mu_{i}$ $$\dfrac{\sum_{i=1}^{n}\lambda_{i}\mu_{i}^{2}}{\sum_{i=1}^{n}\lambda_{i}}\times \dfrac{1}{1+\sum_{i=1}^{n}\mu_{i}^{2}} \...
1
vote
1answer
34 views

Proving there exists a curve whose tangent vector $v$ satisfies $\nabla f \cdot v = 0$

Let $f:\mathbb{R}^3\to \mathbb{R}$ a $C^1$ function, $(x_0,y_0,z_0)\in \mathbb{R}^3$ such that $f(x_0,y_0,z_0)=0$ and $\nabla f(x_0,y_0,z_0)\neq 0$. Let $$S=\{(x,y,z) \ | \ f(x,y,z)=0\}$$ and $v=(v_1,...
0
votes
1answer
50 views

Integral of Continuous and Increasing Function

Suppose that $f: [0,b] \to \mathbb R$ is continuous and increasing and $f(0)=0$. let $g$ denote the inverse of f. Then show that $\int_0^b f(t)dt + \int_0^{f(b)} g(t)dt = f(b)b.$
1
vote
0answers
77 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a self-...
3
votes
1answer
62 views

Breaking a Function in $L^{\infty}[0,1]$

Let $f\in L^{\infty}[0,1]$ s.t. $\|f\|_{\infty}=1$ $E:=\{x\in[0,1]:|f(x)|<1\}$ If $m(E)>0$, then is it possible to find $g,h\in L^{\infty}[0,1]$ such that $\|g\|_{\infty},\|h\|_{\infty}=1$...
10
votes
3answers
149 views

Regularity of the function $|x|^ax$

Assuming $x \in \mathbb{R}$, what can we say about the regularity class ($C, C^1, C^2, ..., \text{or}\ C^\infty$) of the following function (also with respect to $a \in \mathbb{R}$)? $$f(x)=|x|^ax$$
0
votes
2answers
91 views

Show that a function is constant

Let $S$ be a non-empty set of real numbers such that if $a,b$ are distinct elements in $S$, then $|a-b|\geq 1/2014$. Let $f:\mathbb R \to \mathbb R$ be such that the range of $f$ is a subset of $S$. ...
3
votes
1answer
44 views

Fubini type results for Hausdorf dimension?

Suppose that I have a stack of hyperplanes in Euclidean space $\mathbb{R}^n$, let's call each plane $P_{a, x}=\{y\in\mathbb{R}^n\mid \langle y, x\rangle=a\}$ Suppose that a measurable subset $A$ of $\...
0
votes
1answer
17 views

The dimension of the operator if the domain has dimension 2

Suppose $A$ is a linear operator s.t. $A\colon X\rightarrow Y$. If $\dim(X)=2$, what is $\dim(A(X))$?
3
votes
2answers
77 views

“Greatest lower bound function”

If $f $ is a function continuous at $c, h $ is positive and $m$ is a function defined as $ m(h)=\inf \{ f(x): x \in [c,c+h] \}$ , how can I prove that the limit of $ m $ as $ h $ approaches $ 0 $ ...
1
vote
0answers
33 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
3
votes
2answers
89 views

Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
0
votes
1answer
54 views

Open map and adherence

How to prove that $$f :E\rightarrow F ~\text{is open} \Longleftrightarrow f^{-1}(\overline{A})\subset \overline{f^{-1}(A)}, \forall A\subset F$$ where $(E,\tau), (F,\theta)$ are topological spaces. ...
0
votes
1answer
45 views

Prove the following about absolutely convergent complex series

Prove that for every sequence $(a_n)_n$ of complex numbers, if the series $\sum_{n\ge 0} a_n$ is absolutely convergent, then $|\sum_{n\ge 0} a_n| \le \sum_{n \ge 0} |a_n|$. I've been given the ...
6
votes
3answers
133 views

How does Wolfram Alpha come up with the substitution $x = 2\sin u$? Integration/Analysis

I have to integrate $$ \int_0^2 \sqrt{4-x^2} \, dx $$ I looked at the Wolfram Alpha step by step solution, and first thing it does is it substitutes $x = 2\sin(u)\text{ and } \,dx = 2\cos(u)\,du$ ...
2
votes
1answer
109 views

How to compute $\lim_{n\to \infty} n\sin(2\pi n! e)$ [duplicate]

I want to calculate $$\lim_{n\to \infty} n\sin(2\pi n! e)$$ I have used the Stirling approximation and I think the answer is zero . But I think the limit maybe not exists. Can some one help? ...
1
vote
2answers
206 views

Continuous function positive at a point is positive in a neighborhood of that point

Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) > 0$ then there exists a $d > 0$ such that $\forall x$, $f(x) > 0$ with $|x-c| < d$. I can understand ...
2
votes
2answers
152 views

Prove exponent(m)=e^{m}

please show me how to do the third one, I just understand the 1st and 2nd, but i have no idea how to do the 3rd. thank you.
1
vote
1answer
101 views

bounded interval is bounded and connected

Can you please tell me if my proof is correct? Definition: Let $X$ be a subset of $\mathbb R$. We say that $X$ is connected iff the following property is true: whenever $x, y$ are elements in ...
1
vote
0answers
34 views

local maxima of weierstrass function

Does the weierstrass function have uncountably many local maxima on (0,1)? I don't really know how to approach this problem at all, so any help is appreciated
1
vote
2answers
227 views

If $f$ and $g$ are continuous, then max(f, g) is continuous and differentiable

If $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then $\max(f, g)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. I'm asked to either prove or disprove this ...
0
votes
0answers
52 views

Stuck on Induction Proof [duplicate]

Let $0< a_1 < b_1$ and define $a_{n+1} = \sqrt{a_nb_n}$, and $b_{n+1}$ = ${a_n + b_n}\over2$. Use induction to show that $a_n<a_{n+1}<b_{n+1}<b_n$. Also, prove $a_n$ and $b_n$ ...