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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Prove that a set is sigma finite

Let (X, 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 F, B$\subset$A, such that 0 My approach: since $\mu$ has a finite ...
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1answer
7 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 ...
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0answers
15 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 ...
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0answers
7 views

Lebesgue Measure in relation to product measure

Let X=Y=[0,1] and let M be the Lebesgue $\sigma$-algebra on [0,1]. Show that any open subset of X$x$Y is M$x$M measurable. My approach: By the compactness of X$x$Y of every open set in X$x$Y will be ...
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11 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 ...
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0answers
19 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 ...
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2answers
35 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. What can we say ...
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0answers
14 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 ...
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10 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 ...
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1answer
15 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 ...
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1answer
27 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 ...
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3answers
22 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 ...
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1answer
21 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 ...
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2answers
154 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" ...
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1answer
31 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 ...
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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
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1answer
39 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 ...
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2answers
46 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? ...
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1answer
22 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 ...
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1answer
42 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 ...
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3answers
43 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 ...
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0answers
11 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
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2answers
38 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 ...
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2answers
51 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 ...
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2answers
18 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 : $a_n$ ...
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0answers
16 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 ...
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1answer
16 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.
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20 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!
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1answer
20 views

Show uniform continuity $\sum_{n=1}^{\infty}{\cos(nx)(nx)^{1\over 2}\over{(n+1)^2}}.$

Show that the series is uniformly convergent in any interval $[0,a]$ for $a>0$. $$\sum_{n=1}^{\infty}{\cos(nx)(nx)^{1\over 2}\over{(n+1)^2}}.$$ I'm not sure what to do. I think the Weirstrass M ...
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1answer
20 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
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0answers
11 views

Is there a better upper bound for an improper integral?

Let $(a_n)$, $(b_n)$ be the positive sequences such that $a_n\to 0$, $b_n\to\infty$ as $n\to\infty$. In my work, I need to estimate an upper bound for the integral $I_n:=\int_0^\infty {{e^{ - ...
1
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1answer
28 views

Rudin Real & Complex Analysis Thm 3.14

In the proof, the author claims that by Lusin's theorem, $g(x) = s(x)$ except on a set of measure $< \epsilon$ and $|g| \leq \|s\|_\infty$, ($g(x) \in C_c(X)$, $s(x)$ simple and $\mu(\{x:s(x) \neq ...
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1answer
32 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
2
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2answers
60 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
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1answer
30 views

What does it mean that a sequence of functions is bounded in $L^1(I)$?

Let $I = (0, 1)$ and $f_n : I \to \mathbb{R}$ a sequence of functions. What does it mean that $f_n$ is bounded in $L^1(I)$? Does it mean that $$\exists c>0 \;\; \text{such that} \;\; \|f_n\|_1 ...
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44 views

determine $\int x\sqrt{1-x^2}\,dx$

I have to determine $\int x\sqrt{1-x^2}\,dx$ and I have a little question about the substitution. I tried to subsitute $t=1-x^2$. It is $dt=-2xdx$ and therefore $dx=\frac{-dt}{2x}$. But it is the ...
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2answers
41 views

Proving $C \int_0^{\epsilon} \sqrt{\log(\frac{1}{x})} dx \leq C \epsilon \sqrt{\log(\frac{1}{\epsilon})}$. [on hold]

How can I prove the inequality $$C \int_0^{\epsilon} \sqrt{\log(\frac{1}{x})} \,dx \leq C \epsilon \sqrt{\log(\frac{1}{\epsilon})},$$ where $C$ is a constant?
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2answers
36 views

Radius of Convergence of the given series

We need to find the radius of convergence of the series $\sum a(n)x^n,$ where $a(n) = n^{-\sqrt n}$ The ratio test isn't helping..
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22 views

Finf f such that $F \circ F$ is a primitive of f

Find all primitivable functions $f:\mathbb{R} \to \mathbb{R}$ that admits a primitive $F:\mathbb{R} \to \mathbb{R}$ for which $F\circ F$ is a primitive of $f$. From $(F \circ F)'=f$ we get that ...
3
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1answer
90 views

Find f such that $\int_0^x f(t)dt =(f(x))^{2015}+f(x)$

Find all Riemann integrable functions $f:[0,1] \to \mathbb{R}$ such that $\forall x \in [0,1]$ we have $\int_0^x f(t)dt =(f(x))^{2015}+f(x)$. We consider $g:\mathbb{R} \to \mathbb{R}$, ...
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0answers
24 views

Derivative has finite, unequal left and right limits at a point; is the function non-differentiable at this point?

I have a short question, related to the ongoing search of mathematics instructors for counter-examples to common undergraduate mistakes. The classical example of a function that is differentiable ...
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1answer
42 views

How to show $\int_{x_0-\delta}^{x_0+\delta} g(x) > 0$ if $g(x_0)>0$? [on hold]

(a) Let $f$ and $g$ be Riemann integrable functions on $[a,b]$. Prove that if $f(x)\le g(x)$ for all $x\in[a,b]$, then $$\int_a^b f(x) dx \le \int_a^b g(x) dx.$$ (b) Prove that if $g$ is ...
7
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3answers
429 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
0
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1answer
17 views

equivalency of the least upper bound property & convergence of every monotone and bounded sequence in $\mathbb{R}$

I'm aware how to prove convergence of every monotone and bounded sequence in $\mathbb{R}$ by using the completeness of $\mathbb{R}$ (using least upper bound property). But now I want to prove the ...
0
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0answers
11 views

Prove for measures $\mu $and $\nu$ $\nu \perp \mu$ iff $|\nu| \perp \mu$ iff $\nu^+ \perp \mu$ and $\nu^- \perp \mu$

Where $\perp$ means mutually singular. I have a question, as $\nu$ is clearly a signed measure do we assume that $\mu$ is signed or just positive? It follows from $\nu\perp\mu$ with the set in ...
2
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2answers
49 views

The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!)

Update: This is false. See the answers for a counterexample. Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p \right)^{1/p}$$ is ...
1
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1answer
24 views

Why does the substitution $\cos\theta\frac{\partial}{\partial r}-\frac{1}{r}\sin\theta\frac{\partial}{\partial\theta} $ work?

This is an example from my textbook: Function $z(x, y)$ satisfies $$ z_{xx}'' + z_{yy}'' = 0 $$ Let $ x = r\cos\theta $, $ y = r\sin\theta $, and $ w(r, \theta) = z(r\cos\theta, r\sin\theta) $ Then ...
0
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0answers
11 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
0
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0answers
35 views

Common traits of functions which are non-trivial to integrate?

My question is very simple: do there exist certain qualities of functions such that functions which possess these qualities are guaranteed not to have anti-derivatives which are expressable in terms ...
17
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5answers
593 views

What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...