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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1answer
45 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
3
votes
2answers
113 views

Failure of differential notation

Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ...
2
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1answer
39 views

Does the implicit function theorem say something about this? [closed]

Is there a smooth $F:\Bbb{R}^4\to\Bbb{R}$ such that for any open $U\in\Bbb{R}^3$ there is no smooth $g:\Bbb{R}^3\to\Bbb{R}$ with $F(x,g(x))=0$ (provided $F$ takes the value $0$)?
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1answer
216 views

Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy $$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
2
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1answer
70 views

If there exists an integrable function that is not zero a.e., then the measure is $\sigma$-finite

Suppose $f\in L^1(\Omega,\mathcal{A},\mu)$ and $f(x)\neq 0$ for almost every $x\in \Omega$. How to prove $\mu$ is $\sigma-$finite? I only got that $\Omega=\cup_{n=1}^\infty \{x\in \Omega:|f(x)|\geq ...
1
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1answer
79 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
1
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1answer
107 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
3
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3answers
42 views

Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
3
votes
1answer
138 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
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2answers
50 views

Real polynomials, complex zeroes and the Intermediate value theorem

I have a second grad polynomial p(x). For arguments sake lets say $$p(x) = x^2 + 16x + 76$$ I also have an inequation $$p(x) > 0$$ Now the inequation does not have a real solution, but only ...
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0answers
134 views

Is this an immediate consequence of the Straddle Lemma?

As main book, I'm using Bartle and Sherberts "Introduction to Real Analysis". In exercises of section 6.1 it's asked to prove the Straddle Lemma: Let $f:I\rightarrow\mathbb R$ be differentiable ...
3
votes
1answer
166 views

Derivative of the parameter

I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ...
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5answers
601 views

Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$.

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
3
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2answers
330 views

Differentiability of the Cantor Function

I know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] ...
2
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4answers
89 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
4
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0answers
146 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
2
votes
1answer
30 views

Integrability in a domain of $\mathbb{R}^{N}$

Hi everyone: Let $f$ be a function defined on an open set of $\mathbb{R}^{N}$ $(N\geq1)$. Is there any difference between the following two statements? 1) $f$ is locally integrable 2) $f$ admits a ...
2
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1answer
58 views

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
7
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1answer
218 views

Series $\sum \frac{1}{n^2\sin^3n}$

Question : Show that series $\sum \cfrac{1}{n^{2}\sin^{3}n}$ is divergent. Hint: Show that $$\sum \frac{1}{n|\sin(n)|}$$ is divergent. I am interested in other possible proofs for this question. ...
5
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2answers
74 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
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1answer
43 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
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3answers
273 views

Why set of natural numbers is infinite, while each natural number is finite?

In his book Analysis Vol. 1, author Terence Tao argues that while each natural number is finite, the set of natural numbers is infinite (though has not defined what infinite means yet). Using Peano ...
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0answers
92 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
2
votes
2answers
105 views

Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist

Hello I was hoping that someone might be able to verify that the following proof that $\lim_{x\to 0} {1\over x}$ does not exist is correct. First assume that $\lim_{x\to 0} {1\over x} = L$. This ...
2
votes
3answers
127 views

Find the directional derivative of the scalar field

Find the directional derivative of the scalar field: $f(x,y,z)=\log(x^2+y^2+z^2)$ at $P_0(1,1,1)$ in the direction of the straight line $\ P_0P $ where $P=(3,2,1)$ What I have done: ...
1
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1answer
24 views

Rotational invariance and intrinsic properties of the measure.

Suppose that we have $v\in\Bbb R^3$ and we have a smooth function $s=s(|v|)$ such that $|v|^4\in L^1(s(|v|)dv)$. It is easy to show (by passing to polar coordinates) that $$\int_{\Bbb ...
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0answers
35 views

Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
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3answers
393 views

Infimum and supremum of a set between 0 and 1

I am a little confused on what the infimum and supremum would be for the set S of all rational number between (0,1) not including 0 and 1. If 0 and 1 were included the answer is quite obvious.
3
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2answers
69 views

How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$

How to prove $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$$. I try to do like $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{n+m=N}^\infty ...
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0answers
52 views

problem of computing limit

The problem is to prove the following for $n \geq 3$ $$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ...
1
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1answer
77 views

Pointwise and uniform convergence of sequence of functions

Let $(f_n)$ be a sequence of continuous functions on $\mathbb R$. If $(f_n)$ converges to $f$ pointwise on $\mathbb R$, then $$\lim\limits_{n\to ...
1
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1answer
44 views

How to prove $\int_e f_k(x)dx=0$ for $k=1,2,\cdots,n$

Let $f_k\in L(E),(k=1,2,\cdots,n)$. If $0<m(E)<+\infty, \ \ \int_E f_k(x)dx=0, k=1,2,\cdots,n.$ Then how to prove for any $\lambda:0<\lambda<1$, there is a subset $e: m(e)=\lambda m(E)$ ...
0
votes
1answer
55 views

Uniform continuity and limit

This is related to my other question. Consider the function $a(s)=\dfrac{1}{1+s^2}$. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous. How ...
0
votes
1answer
114 views

Partition of Unity

Suppose $K \subset \mathbb{R}^n$ be compact and $\{s_j\}_{j=1}^\infty$ be a countable dense set of $K$. Define, $u_s(x) = \max\{2-\frac{|x-s|}{dist(x,K)},0\}$, $\sigma(x) = \sum_{j=1}^\infty 2^{-j} ...
2
votes
1answer
85 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
0
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2answers
76 views

Limit of an integral, as the measure of the region of integration approaches zero

Hi everyone: Let $f$ be a function defined on on open set $D$ of $\mathbb{R}^{N}$, $(n\geq1)$. Suppose that $(\Omega_{\varepsilon})$ is a family of measurable sets in $D$ such that ...
2
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3answers
111 views

Prove $\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$ using Dominated or Monotone Convergence

Is there a way to prove that $$\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$$ via the Dominated or Monotone Convergence Theorem?
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1answer
112 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
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0answers
91 views

About an example

I want to find a function $f$ such that $|f(x)|\leq k |x|^{\alpha} $ where $k>0$ and $\alpha\in[0,1)$ for all $x\in \mathbb{R}$ $f'(0)=0$ Please thank you.
3
votes
1answer
86 views

Uniform continuity and translation invariance

Consider the function $a(s)=\dfrac{1}{1+s^2}$ and the space $X=\{f:\mathbb{R}\to \mathbb{R}$ such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous$ \}$. I want to show that $X$ is ...
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1answer
132 views

Prove that the a modified Cantor Set is not Jordan-Measurable

Let $C_0 = [0,1]$ and if $C_n$ is given as a disjoint union of intervals, construct $C_{n+1}$ by removing from each interval $I$ an open interval of length $(n+2)^{-2}|I|$ in the middle of each ...
2
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0answers
32 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
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2answers
62 views

f(x,y) jointly differentiable

What is the definition of "jointly continuously differentiable function"? I.e. when $f:\mathbb{R}^2\rightarrow \mathbb{R}:(x,y)\mapsto z$ is jointly continuously differentiable in $x,y$ ? Is it ...
1
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1answer
83 views

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Let $F \subset \mathbb{R}$ be a closed set and define the distance from $x \in \mathbb{R}$ to $F$ by $d(x,F)= \inf_{y \in F} |x−y|.$ Prove that $$\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$$ for a.e. ...
7
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1answer
311 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
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0answers
31 views

Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and $\lambda\leq \sigma_1, \sigma_2 \leq ...
0
votes
1answer
134 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
2
votes
3answers
57 views

Uniform and point wise convergent

What is the difference between point wise convergence and uniform convergence? Can you explain the answer geometrically?
4
votes
2answers
300 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
1
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1answer
41 views

Which step of this proof needs more justification? (just a sketch)

I want to show that $T(f) = \int_{0}^{1} |f|^2 dx$ is continuous where $T: C[0,1] \to \mathbb{R}.$ Basically I know that $f(x)^2 - f(a)^2 = (f(x) - f(a) )(f(x) + f(a) ) < \epsilon M.$ I bound ...