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.

learn more… | top users | synonyms

0
votes
0answers
15 views

Lebesgue measure of union of disjoint measurable sets

I am wondering if the Lebesgue measure of the union of a countable collection of disjoint measurable sets is equal to the sum of the measure of such sets. I feel that they may be equal. I know that ...
1
vote
3answers
36 views

Prove that this set is open

$$ A=\left\{ x \in \mathbb{R}^{p} \mid \forall i:\ x_{i} \in (-1,1) \right\}$$ Pick $x\in A$ at random, and choose $\delta = \min B$ where: $$ B = \{ 1-x_{i}\mid i \in \{1,2,\dotsc,p\} \} \cup \{ ...
0
votes
4answers
47 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
0
votes
1answer
14 views

Continuity of $f$ on $I$ where $f_n$ is continuous on $I$ and it converges uniformly to $f$ on $I$

Let $I=[a,b]$ be a bounded and closed interval, let $f_n$ be a sequence of functions on $I$ and $f:I\rightarrow\Bbb{R}$. $f_n$ is continuous on $I$ for all $n\in\Bbb{N}$ and it converges uniformly to ...
0
votes
2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
0
votes
1answer
26 views

Evaluate $\lim _{n\to \infty }\left(1-f\left(\frac{1}{\sqrt{n}}\right)\right)\cdot \sum _{k=1}^nf\left(\frac{k}{n}\right)$

We have $f:\left[0,\frac{\pi }{2}\right]\rightarrow R,\:f\left(x\right)=cos\left(x\right)$, and we need to evaluate: $\lim _{n\to \infty ...
0
votes
0answers
7 views

Kernel estimate in boundary point

Good moorning, I wonder how to prove that if $X_{1}, \ldots, X_{n}$ are iid from exponential distribution with expected value 1, then the expected value of its kernel density estimator in zero is ...
-1
votes
1answer
26 views

Translation property in $L^1(\mathbb{R})$ space

Let $g(x)$ be a bounded measurable functions on $\mathbb{R}$, and $f(x)$ be in $L^1(\mathbb{R})$. Notation: $\int_\mathbb{R} h(x)dx=\ $the integration of measurable function $h$ over $\mathbb{R}$ I ...
-2
votes
2answers
52 views

How do I show that $0<a_n^2<a_n$ If $\sum _{n=1}^\infty a_n$ is convergent?

Since $\sum _{n=1}^\infty a_n$ converges, i know know that $\lim _{n\to \infty}a_n=0$. I know I have to use the comparison test to show that ${a_n}^2$ converges but how do i show that ...
4
votes
1answer
94 views

Where is Cauchy's wrong proof?

Allegedly, Cauchy mistakingly "proved" that pointwise convergence of continuous functions is continuous. I saw this somewhere in a book, and it is also in wikipedia: Uniform convergence. In his ...
1
vote
0answers
53 views

Shifting to polar coordinates

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in ...
-1
votes
2answers
47 views

Power Series Problem… [on hold]

If $$\sum_{n=0}^\infty a_n x^n=0$$ does that imply $a_n$ are equal to zero?
2
votes
1answer
34 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
-2
votes
1answer
32 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
0
votes
1answer
30 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
-3
votes
1answer
37 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
0
votes
1answer
24 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
2
votes
0answers
55 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
1
vote
2answers
46 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
1
vote
1answer
30 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
1
vote
2answers
40 views

Prove that $\int_0^1\int_x^1 \frac{f(y)}ydy\,dx=\int_0^1f(x)\,dx$ if $f$ is Lebesgue integrable

Suppose f is a Lebesgue integrable function on [0,1] and define a new function by $$g(x)= \int_x^1 \frac{f(y)}ydy$$ for all x in [0,1]. Prove that $$\int_0^1{g(x)} dx=\int_0^1{f(x)} dx$$ My ...
1
vote
1answer
30 views

Prove that a set is sigma finite

Let $(X, \mathcal{F}, \mu)$ be a $\sigma$-finite measure space and let $A$ in $F$ be such that $\mu(A)=\infty$. Prove that there exists $B$ in $ \mathcal{F}$, $B\subset A$, such that $0< \mu(B)< ...
3
votes
1answer
17 views

Problem with real differentiable function involving both Mean Value Theorem and Intermediate Value Theorem

Problem: Let $a,b \in \Bbb R$, $a<b$, and let $f$ be a differentiable real-valued function on an open subset of $\Bbb R$ that contains $[a,b]$. Show that if $\gamma$ is any real number between ...
0
votes
0answers
24 views

Pointwise and uniform convergence of increasing functions

Let $a< b$ and assume $f_n : [a,b] \to \Bbb R$ are increasing functions, $ n = 1,2,\dots.$ Prove that if $f_n \to f$ pointwise on $[a,b]$, then (i) $f$ is increasing, and (ii) if $f$ is continuous ...
0
votes
0answers
22 views

Lebesgue Measure in relation to product measure

Let $X=Y=[0,1]$ and let $\mathscr M$ be the Lebesgue $\sigma$-algebra on $[0,1]$. Show that any open subset of $X\times Y$ is $\mathscr M\times \mathscr M$ measurable. My approach: By the compactness ...
2
votes
1answer
51 views

a question about differential geometry(Gauss-bonnet theorem and isolated singular point in the surface)

Let C be a regular closed simple curve on a sphere $S^2$. Let v be a differentiable vector field on $S^2$ such that the trajectories of v are never tangent to C. prove that each of the two regions ...
1
vote
0answers
27 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
4
votes
3answers
66 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
0
votes
0answers
18 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
1
vote
0answers
19 views

convergence of a numerical method

given a function $f:\mathbb R\to\mathbb R$ in of class $C^3$. We suppose that there exists $s\in \mathbb R$ such that $f(s)=0$ and $f'(s)\neq 0$. Let $\beta$ be a real number s.t. $\beta \neq 0$. We ...
0
votes
1answer
17 views

Differentiability functions

If $f:A\subset \mathbb R^n\rightarrow \mathbb R^m$ and $g:B\subset \mathbb R^n\rightarrow \mathbb R^m$ are differentiable functions on the open sets A, B and $\alpha,\beta$ are constants. Prove that ...
1
vote
1answer
36 views

How to show that every set with Lebesgue outer measure zero is Lebesgue measurable?

Definition of Lebesgue measurable if for each $ε>0$, there exist a closed set $F$ and an open set $G$ with $F⊂E⊂G$ such that $m$ * $(G-F)<ε$. About this problem, $F$ can be a empty set that is ...
0
votes
3answers
24 views

Borel $\sigma$-algebra defintion question

So I am studying measure theory and I have found myself struggling to fully understand the concept of the Borel $\sigma$-algebra in depth. We know that the Borel $\sigma$-algebra is the smallest ...
0
votes
1answer
25 views

Difficulty with a differentiation of measures proof

This shows up in a proof about differentiating measures. I'm having trouble figuring it out: For any $x \in \mathbb{R}^n$, let $\mathcal{C}_r(x)$ denote the set of open cubes with diameter less than ...
5
votes
2answers
175 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
1
vote
1answer
41 views

a question about undergraduate-level differential geometry(Gauss-Bonnet theorem)

Let $S\subset R^3$ be a regular surface homeomorphic to a sphere. Let $\alpha\subset S $ be a simple closed geodesic in S,let A and B be a regions of S which have $\alpha$ as a common boundary. Let ...
0
votes
0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
4
votes
1answer
50 views

Convergence of $ L^{p} $-integrals implies convergence in $ L^{p} $-norm?

Let E be a measurable set, $\{ f_n \}$ and $f$ are in $L^p(E)$ such that $f_n \to f$ pointwise a.e. If $\lim \|f_n \|_p = \| f \|_p$, is it true that $\lim \| f_n - f \|_p = 0$? I have tried ...
2
votes
2answers
51 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
0
votes
1answer
25 views

Condition for convergence almost uniformly

Let $(X,\Sigma,\mu)$ be a measure space. $(f_n)$ and $f$ measurable functions and, for $\epsilon >0$ and $k\in\mathbb{N}$: $$D_k(\epsilon):=\{x\in X: |f_k(x)-f(x)|\ge\epsilon\}.$$ We suppose that ...
1
vote
1answer
48 views

Using stokes' theorem

$B=\{(x,y), x^2+y^2\le1\} $ is a closed ball and $S=\{(x,y,z), z=x^2+y^2, (x,y)\in B\} $ oriented so that $f:B\to S$ defined by $$f(x,y)=(x,y,x^2+y^2)$$ is orientation preserving. Compute ...
2
votes
3answers
52 views

Continuity Rewritten: $\forall\delta>0,\exists\varepsilon>0\dots$

Take the following definition of the continuity of a function $f$ at some point $x_0$: $$ \forall\varepsilon>0,\exists\delta>0\text{ such that ...
0
votes
0answers
16 views

“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
3
votes
2answers
47 views

Upper and Lower Darboux integral of a piecewise function $f(x)=x$ and $f(x)=0$.

Let $0<a<b$. Find the upper and lower Darboux integrals for the function $$f(x)=x$$ if $x\in[a,b]\cap\mathbb{Q}$ and $$f(x)=0$$ if $x\in[a,b]-\mathbb{Q}$. I am so lost on this problem. Any ...
1
vote
2answers
55 views

Fundamental Theorem of Calculus, application

I want to derive the function $$F(x)=\int_a^{x^2}\sin^3t\,dt$$ with the fundamental theorem of calculus, but I dont know how to handle the $x^2$. Maybe with subsitution I think Fundamental theorem ...
-1
votes
2answers
22 views

Radius of Convergence for a given sequence of functions

If $$\begin{cases} a_n = 1\ \text{ if } \exists k\in\mathbb{N}\ n=k^2\\ a_n = 0\ \text{ otherwise}\end{cases}$$ find the radius of convergence of $\sum_{n=0}^{+\infty}a_n x^n$. What i tried : $a_n$ ...
-2
votes
0answers
34 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
0
votes
0answers
18 views

Continuity of optimisation problem

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$ I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...
0
votes
1answer
27 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
0
votes
2answers
22 views

Does a.e. convergence imply the boundness in $L^1$?

Let $f_n : I = (0, 1) \to \mathbb{R}$ be a sequence of functions. If $$f_n \to 0 \;\; a.e$$ does it imply that $$f_n \;\; \text{is bounded in} \;\; L^1(I)?$$ Why yes/not? Thank you!