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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Upper integral of a production funcion, can i assume this about the supremum?

Let $f: A \rightarrow \mathbb{R}$ $g: B \rightarrow \mathbb{R}$ be bounded and non negative functions at blocks $A$ and $B$. Define $\phi: A\times B \rightarrow \mathbb{R}$ as $\phi(x,y) = f(x)g(y)$. ...
2
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1answer
24 views

$f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
-1
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0answers
12 views

Continuously differentiable function injective on convex set

Can you help me solve the following exercise: (a) Let $n\in \mathbb N$ and $G \subset \mathbb R^n$ a convex set, $f:G\to \mathbb R^n$ continuously differentiable with $$det(J_f(c_n)) \neq 0 \...
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2answers
18 views

How to prove differetiability in $\Bbb K^2$?

I have to investigate differentiability in all points of the following function: $$f: \Bbb {R}^2 \to \Bbb R \: \: \: \: \: \: \: f(x,y):=\begin{cases} y-x &\mbox{if } y\ge x^2 \\ 0 & \mbox{if }...
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0answers
24 views

Show that integral is analytic

Let $h:[0,\infty)$ be an integrable function. Prove that the function $$g(z)=\int_0^\infty h(t)e^{tz}\,dt$$ is analytic on $\{z=x+yi:x<0,y\in\mathbb{R}\}$. How do I start for this question? I ...
1
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1answer
22 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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0answers
27 views

Let $f:\mathbb R^n\to\mathbb R^n$ of class $C^1,\;f(x)=0\quad\forall\,\lvert x\rvert\geq r>0$ such that $\displaystyle\int_{B[0,k]}\det J(f(x))=0$ [on hold]

Let $\;f:\mathbb{R}^n \to \mathbb{R}^n$ of class $\,C^{1}$, exists $\,r>0\,$ such that $\,f(x)=0\,$ for any $\,\left\lvert x \right\rvert \geq r$. Prove that there exists $\,k>0\,$ such that $$\...
2
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2answers
82 views

Rebracketing Theorem

My questions regarding the below theorem Both questions are centred on Eq(2) and the paragraph preceding it. 1) How is it that Eq(2) contains $a_k$ but in that section of the proof the assumption is ...
4
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1answer
480 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point $x_0$...
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45 views

Question regarding continuity ($\epsilon$-$\delta$)

In this post: Proof continuity of a function with epsilon-delta people explained how to proof that $$f(x) = \frac{x-1}{x^2+1}$$ is continuous in $x_0 = -1$. However, my solution is the following: $$|f(...
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1answer
45 views

$f: \mathbb R^2 \to \mathbb R$ is differentiable when both partials exist and one is continuous

I'm trying to solve the following exercise: Let $f:\mathbb R^2 \to \mathbb R$ a continuous function whose partials exist everywhere in $\mathbb R^2$. Show that $f$ is ($\mathbb R$-)differentiable ...
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2answers
92 views

would changing the lower limit of a power series affect radius of convergence

When we change the lower limit of a power series by any finite quantity, would it increase or decrease radius of convergence or no change? Clarification of terminology: There might be confusion about ...
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2answers
80 views

On the continuity of $f(x,y)=g(x)\cdot h(y)$

Let $g:A \subset \mathbb{R} \to \mathbb{R}$ and $h:B \subset \mathbb{R} \to \mathbb{R}$ (non identically null). Consider $f:A \times B \to \mathbb{R}$ such that $$f(x,y)=g(x)\cdot h(y).$$ Let $(x_0,...
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0answers
6 views

The peano curve mapping- finite and infinite ternary's.

Peano defined a map $f_p$ from a unit interval to the unit square in terms of the operator $$kt_j=2-t_j$$ where $(t_j=0,1,2) $ as $$f_p(0_3.t_1t_2t_3t_4...)=\left(\begin{array}{c} 0_3.t_1(k^{t_{2}}t_3)...
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0answers
31 views

Taylor Series coefficients

I have performed a Taylor Series expansion of a 2-D function in variables (y1,y2) and got something like: $f(y_1,y_2) = 3y_2 + 0.5y_1^2 + ...$ My question is that I would like this to "match" to ...
2
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1answer
76 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
0
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1answer
12 views

In Constrained Optimization, Restrict Domain to Open Set $A\subset\mathbb{R^N}$?

In constrained optimization and context of economics (e.g. utility function with quantity of goods as arguments subject to wealth), why do textbooks always restrict domain of the objective function ...
2
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1answer
37 views

Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
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1answer
27 views

Integral substitution

I don't understand why the integral boundary change here from $[0,1]$ to $[0,\infty]$ $$\int_0^1 \int_{0}^\infty xe^{-x}f(ux,(1-u)x)\mathrm{d}u \mathrm{d}x$$ Substitution: $(ux=t,\ (1-u)x=s)\implies ...
3
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2answers
60 views

What to know about convergence of integrals

According to the values of p>0 examine the convergence of the integral: $$\int_0^{+\infty} \dfrac{\ln(1+2x^{3p})}{(x+x^2)^{4p}\arctan(x)^{1/2}}dx$$ I didn't find a good explanation about this kind of ...
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2answers
26 views

Lebesgue integral question (double integral)

Let $g,h$ be nonnegative Lebesgue measurable functions on $\mathbb{R}$. Prove that $$\int_{-\infty}^\infty g(x)^2h(x)\,dx=\int_0^\infty\int_{\{t\in\mathbb{R}:g(t)>x\}}2h(t)x\,dtdx.$$ I am lost on ...
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9answers
4k views

Proof that the set of irrational numbers is dense in reals

I'm being asked to prove that the set of irrational number is dense in the real numbers. While I do understand the general idea of the proof: Given an interval (x,y) choose a positive rational ...
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3answers
37 views

Question about proving infinite union of sequence of countable sets is countable.

Let {$E_n$} , $n$=1,2,3... be a sequence of countable sets, and then $S$ = $\bigcup_{n=1}^{\infty}$ $E_n$ The proof in my book says Let every $E_n$ be arranged in a sequence {$x_{nk}$} $k$=1,2,3.....
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1answer
39 views

Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
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0answers
24 views

Difference of two continuously differentiable functions is also continuously differentiable?

If f is defined as $f_1-f_2$, and $f_1,f_2$ are both continuously differentiable. Can I say that f is also? If not, what other conditions do I need, maybe the boundedness of f1 and f2? Also, can ...
3
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2answers
61 views

Prove that the integral $\int_0^1 f(t) P(t - x)dt$ is a polynomial in $x$

So suppose $f$ is a generally complex, continuous function on $[0,1]$ and $P$ is a polynomial defined on the real numbers. Evidently $$ \int_0^1 f(t)P(t - x)\,dt $$ is a polynomial in $x$ but that ...
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3answers
33 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...
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3answers
47 views

Problem proving “p”, “k” are limit point.

Good night, i have problem with this exercises: $A=\left\{ 1-\frac{1}{n}:n=1,2,\ldots\right\} $ I make this: $$ \lim_{n\rightarrow\infty} \left(1-\frac{1}{n}\right)=1$$ Prove: Be $p=1$ and $r>...
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1answer
79 views

Proof of the Arzelà–Ascoli Theorem

I'm stuck on a particular line of the proof of The Arzelà–Ascoli Theorem. In lectures, we have: $1.$ Defined equicontinuous as: Let $X$ be a metric space, $C(X) = \{f: X \rightarrow \mathbb{R}\...
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1answer
34 views

Trigonometric substitution [illustration / right triangle derivation]

Real quick: If I have the function $$\int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$$ I can easily substitute by setting $x$ equal to $a\sin \theta$. But why actually is that? If I draw a right ...
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1answer
91 views
+100

Integrability question with a function on a box in $\mathbb{R}^2$ (bounty added)

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
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1answer
238 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
3
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2answers
55 views

Analysis Problem [on hold]

Given that: For any $[a, b]\subset (-\infty,+\infty)$, $f$ is integrable in $[a,b]$, $p>0$, and ${\mid f\mid}^{p}$ is integrable in $(-\infty,+\infty)$. Prove that $$\lim_{h\to0}\int_{-\infty}^{+\...
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0answers
20 views

Alternative Proof about Product Measures and Iterated Integrals

Background Theorem 2.36 of Folland's Real Analysis says that if $(X,M,\mu)$ and $(Y,N,\nu)$ are sigma finite measure spaces, and $E\in M\bigotimes N$, then $x\mapsto \nu(E_x)$ and $y\mapsto \mu(E^y)$ ...
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5answers
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Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is "...
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1answer
28 views

$f:\mathbb{R^N}\rightarrow\mathbb{R}$ Definition of Partial Derivative Using Limit or Epsilon

Can someone share the exact definition of partial derivative for a function $f:\mathbb{R^N}\rightarrow\mathbb{R}$ in both limit language and epsilon-delta language? In particular, I have hard time ...
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1answer
52 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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2answers
596 views

Show that $f^*(x) = \sup \{ f(y) : a \leq y \leq x \}$ is a non-decreasing continuous function

I am currently working on a problem and stuck on it. Here is the problem (it comes form Elementary analysis, the theory of Calculus by K. Ross P.153): Q: Let $f$ be a continuous function on [a,b]. ...
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1answer
12 views

Derivative of volume of given set

As picture below ,how to compute the $\partial_t |\Omega_t|$ ? The picture below is from the 32 page of Maximum principles and the method of moving planes. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
38 views

A Bending Buzz Wire Game

There is a wire connecting an exit and an entry point. At the entry, the wire has height $0$, at the exit, it has height $1$. Since the wire is connected, the wire has height $1/2$ somewhere, whatever ...
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2answers
39 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
2
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1answer
145 views

Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.

I really need help with this question: Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.
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0answers
17 views

Let $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ given by $\phi(t)=A(tx)$, then $\phi'(t). h=(A'(tx). x). h$ or $(A'(tx). h). x$?

Let $U$ be an open ball centered in $0$ in $\mathbb{R}^m$. Given $\phi:(0,1)\to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ be defined by $\phi(t)=A(tx),$ where $A:U\to \mathbb{R}^n$ and $x\in U$, which ...
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1answer
44 views

Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
4
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1answer
42 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
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0answers
18 views

Question about $N-$functions

If $\Phi$ is a $N-$function, and $(u_n)$a sequence from $W^{1,\Phi}(\mathbb{R}^N)$ such that $u_n\rightarrow u$ in $W^{1,\Phi}(\mathbb{R}^N)$ Can we prove that $$\Phi(|u_n|)\leq \Phi(|u_n-u|+|u|)$$ ...
1
vote
1answer
19 views

Proof radius of convergence for zero and infinite (power series)

I deleted my last question because there was a huge mistake inside. Given: $R$ is the radius of convergence of $\sum_{n=0}^{\infty} a_{n}x^{n}$, also suppose that $\lim_{n\rightarrow \infty} \left | \...
3
votes
1answer
29 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
3
votes
1answer
39 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
0
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2answers
55 views

A paradox on countability.

Let, $\mathbb{P}$ be the set of prime numbers. We know that $\mathbb{P}$ has a bijection with set of Naturals $\mathbb{N}$. That is, $\mathbb{P} \stackrel{}{\longleftrightarrow} \mathbb{N}$. Again, ...