Questions tagged [real-analysis]
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
143,427
questions
-1
votes
0
answers
23
views
Decide if f is Riemann integrable on [0,1] if the answer is affirmative, compute f(t)dt. See image attached.
Here is my progress, but not sure if the double summation is correct.
1
vote
0
answers
35
views
Is every continuous bijection from $\mathbb{R}$ to itself of finite even order an involution?
The following question is inspired by Michael Penn's YouTube video Functions that "cube" to one..
Given any continuous bijection $f:\mathbb{R} \to \mathbb{R}$ for which the order of $f$ in ...
- 9,533
-1
votes
0
answers
20
views
Approximation in outer measure
Suppose that for any set $E$ in the product ${\sigma}-$algebra ${{\mathcal F}_A := \prod_{i \in A} {\mathcal F}_i}$ on ${\Omega_A := \prod_{i \in A} \Omega_i}$, there exists an elementary set ${E_\...
- 211
0
votes
0
answers
11
views
Proving a criteria for weak compactness of Radon measures
I'm trying to prove the following criteria for weak compactness of Radon measures:
Let $\{\mu_k\}_{k \in \mathbb{N}}$ be a sequence of radon measures on $\mathbb{R}^n$ satisfying $$\sup_{k \in \...
- 6,667
2
votes
0
answers
49
views
Exploring the Converse of the Stolz-Cesàro Theorem: Necessary Conditions and Existence.
Stolz-Cesàro theorem case $\frac{\infty}{\infty}$:-
If $b_n $ is a monotone increasing sequence and $\lim \limits_{n \to \infty} b_n = \infty $,
and if $\lim \limits_{n \to \infty} \frac{a_{n+1}-a_n}{...
- 2,216
0
votes
0
answers
23
views
Pseudo Thomae's function
Let $f: \mathbb{R^{3}} \to \mathbb{R}$ with
$f(x,y,z)$ =\begin{cases}
\frac{1}{q},&\text{if $(x,y,z)=(\frac{p}{q},\frac{r}{l},\frac{m}{n})$ with $p,r,m\in\mathbb{Z}$ and $q,l,n\in\mathbb{Z}\...
5
votes
2
answers
74
views
Convergence analysis of a quotient of two sequences $x_{n+1}^2 = x_n^2 + \frac{c}{x_n^2}$.
Given two recurrence relations
$x_{n+1}^2 = x_n^2 + \frac{c_1^2}{x_n^2}$ and $y_{n+1}^2 = y_n^2 + \frac{c_2^2}{y_n^2}$ for $c_1, c_2\in \mathbb{R}^+$ and $x_1=y_1=c_3\in\mathbb{R}^+$.
We want to study ...
- 59
-1
votes
1
answer
43
views
Directly proving a metric function is continuous
Let $(X,d)$ be a metric space an $p\in X$ be any point. I want to show that a function $f:X\to\mathbb R$ defined by $f(x)=d(x,p)$ is continuous.
My attempt:
Let $G\subseteq\mathbb R$ be open and ...
- 271
3
votes
2
answers
59
views
If $g(t,x) \rightarrow 0$ as $x \rightarrow \infty$, then decays uniformly in $t$?
Assume we have continuous $g:[0,T]\times \mathbb{R}\rightarrow \mathbb{R}$ such that
$$
\lim_{x \rightarrow \pm \infty} g(t,x)=0
$$
for every $t \in [0,T]$.
Then let $(t_k)_{k\in\mathbb{N}}\subseteq [...
- 6,136
1
vote
0
answers
22
views
Show that image of {g'(x) = 0} has measure zero for absolutely continuous functions [closed]
How would you prove the following statement:
For $g: [a, b] \rightarrow \mathbb{R}$ an absolutely continuous function, show that for $E = \{x : g'(x) = 0\}$, we have $m[g(E)] = 0$ (the Lebesgue ...
- 114
3
votes
1
answer
83
views
how to solve $m$ in inequality involving $m \log m $?
Let $m$ and $n$ be positive integers. I am trying to prove that if $m \log m >0.53n$, then
$$m>\frac{0.53n}{\log 0.53n},$$
where $m$ is not necessarily bigger than $n$.
My attempt:
First recall ...
- 572
0
votes
0
answers
11
views
Proof verification: if $\Omega$ is a convex subset of a topological vector space $X$, then $\overline{\Omega}$ is convex.
I'm working on a proof that if $\Omega$ is a convex subset of a topological vector space $X$, then so is its closure, $\overline{\Omega}$. My proof was different than the proof in the book, so I want ...
- 967
1
vote
1
answer
39
views
Is my proof for the statement, "Prove that a continuous function on a closed, bounded interval is bounded", correct?
There are 2 parts to this question:
Let $f$ be continuous on the closed interval $[a, b]$. Then $f$ is bounded on $[a, b]$.
Assume for contradiction that $f$ is unbounded. For any $M \in \mathbb{N} &...
- 85
0
votes
0
answers
14
views
Uniform continuity of $h$ satisfying $ h(a,y) \geq \alpha$ for al $y$ and $a$ is fixed
Let $h:\mathbb{R}^d \times \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ mapping and $a\in \mathbb{R}^d$. Assume that there is $\alpha>0$ such that $$ h(a,y) \geq \alpha,$$
for all $y\in \mathbb{R}^n$.
...
2
votes
3
answers
159
views
Proving an inequality of a function given its derivative's values
Let $f(x)$ be differentiable twice and for which it's true that $f(0) = f'(0) = 1$ and $f''(x) > \frac{1}{x^2 + 1}$ for all values of $x$. Prove that $\forall x \ge 1: f(x) > 2$.
I thought this ...
- 119
0
votes
1
answer
67
views
Prove and determine all values of p for which the sequence below converges, in light of the conditions given.
How do I prove the following:
Let p > 0 be a real number and let (a_n) be a sequence of positive real numbers s.t. lim as n approaches infinity n^p(a_n) ^1/2 = 1. Prove and determine all values of ...
1
vote
0
answers
21
views
Behaviour of the solution of a non-local ODE
I am studying the following non-local ODE in $f$
$$\dot f(x) \left\{\log(x + \varepsilon) - 2\log(f(x))\right\} + \int_{x}^{2x_0} \frac{\dot f(s)}{s + \varepsilon}ds = c$$
for some constant $c \in \...
- 3,923
1
vote
0
answers
50
views
Orthogonal projection on the null space of $L$: Where does the minus come from?
I am referring to this paper.
For $F=(f,g)^\top$, and the linear operator
$$
\begin{equation*}
LF=\begin{pmatrix}
-\rho_\infty\chi_1\rho_g-\frac{1}{\rho_{\infty}} f \\ -\frac{1}{\rho_\infty}\chi_2\...
- 2,017
0
votes
0
answers
14
views
Well-posedness of Fredholm integral equation of the second kind
In several papers and other sources, I have seen statements about it being `well-known' that the Fredholm integral equation of the second kind is well-posed, in contrast to a Fredholm integral ...
- 3,013
-3
votes
0
answers
50
views
$1+\left(\frac{2}{3}\right)^{2}+\left(\frac{2\cdot4}{3\cdot5}\right)^{2}+\left(\frac{2\cdot4\cdot6}{3\cdot5\cdot7}\right)^{2}+...$
$$1+\left(\frac{2}{3}\right)^{2}+\left(\frac{2\cdot4}{3\cdot5}\right)^{2}+\left(\frac{2\cdot4\cdot6}{3\cdot5\cdot7}\right)^{2}+...$$
Any hints on the above series. here both root and ratio test gives ...
- 149
2
votes
2
answers
67
views
is $A=\{v|v_i\leq v_{i+1},\forall i\in\{1,2...,n-1\}\}\subset\mathbb{R}^n$ clopen in $\mathbb{R}^n$
The original question is
Let $A=\{v|v_i\leq v_{i+1},\forall i\in\{1,2...,n-1\}\}\subset\mathbb{R}^n$. Show that $A$ is a closed set in $\mathbb{R}^n$
Some idea:
Method 1: This is pretty straight ...
- 1,230
2
votes
0
answers
44
views
Question regarding Mean value theorem and a monotonically increasing function $f'$
I was given the following information about a function:
Let $f$: [0,$\infty$) be a function so that
$f(0)=0$
$f$ is continuous on [0,$\infty$)
$f$ is differentiable on (0,$\infty$)
the function $f'$: ...
1
vote
0
answers
27
views
ODE for mean of SDE
Given a stochastic differential equation
$$ dX_t = f(t,X_t) dt + \sigma(t,X_t)dB_t$$
where $B$ represents standard Brownian motion, if I wanted an ODE for the mean of the SDE, I could write for any $T ...
- 1,074
3
votes
0
answers
85
views
Can you critique my exposition of $\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$?
Prove
$$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$$
using Fubini's Theorem.
My solution is below. Proof is Correct.
What I want to know is :
Is it well written?
How could the writing be improved, ...
- 2,830
5
votes
3
answers
119
views
For which values of $\alpha \in \mathbb{R}$ does $\sum_{n=1}^{\infty}n^\alpha \left (\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}} \right )$ converge
I have a question in my analysis course that I have been trying for a long time and I'm not quite sure how to answer it. I have an idea for an answer but I'm not sure if it's right.
Also note the only ...
- 135
1
vote
1
answer
65
views
Differentiable function satisfying $x \cdot \nabla f(x) = 0$
We have a differentiable function $f: \mathbb{R}^n \setminus \{0\} \rightarrow \mathbb{R}$ that satisfies
$$
x \cdot \nabla f(x) = 0 \quad \forall x \in \mathbb{R}^n \setminus \{0\}
$$
I am trying to ...
- 308
0
votes
0
answers
10
views
Requirements for convexity of isolated minima
Let $\mathcal{F} \in \mathscr{C}^1(X, \mathbb{R}^+ \cup {0})$, where $X$ is a separable Hilbert space. Consider any isolated minimum $\phi \in X$ ($\phi$ has a non-zero neighborhood in which it is the ...
- 1
4
votes
1
answer
83
views
How to prove a result about the accumulation point / cluster point in Brownian Motion?
I have a standard Brownian motion, $B(t)$, $t\ge0$.
I am trying to prove the following result:
Every $t > 0$ is an accumulation point (i. e., cluster point) of $(s: B(s) = B(t))$ from the right, ...
- 215
2
votes
1
answer
29
views
convergence of a numerical series using information about an entire series
I'm on a problem that seems simple but turns out to be a bit twisted.
Let be $\sum_{n\epsilon N }^{}{u_nz^n}$ a power series with radius of convergence ρ = 1. Which of the following statements are ...
Zak
1
vote
0
answers
37
views
Understanding $H(x,y) = \dfrac{\int_x^y f(t)dt}{\int_x^y g(t)dt}$
Let $f, g : [0,\infty) \to \Bbb R^+$ be two positive and measurable functions. Define $$H(x,y) = \dfrac{\int_x^y f(t)dt}{\int_x^y g(t)dt}.$$
I'm trying to gain understanding about what this function $...
- 11
-2
votes
0
answers
18
views
Convergence in $L^p$ of a function $u_k$ implies convergence almost everywhere for a subsequence.
I am trying to learn measure theory by my own and in one part of the book I am reading (Measures, Integrals and Martingales by Schilling) I am trying to understand how can I prove the statement above:
...
3
votes
0
answers
122
views
$\lim\limits_{x\to+\infty}\sum_{k=0}^\infty \left[\lambda f\left(\frac{x}{2^k}\right)-f\left(\frac{x}{3\cdot 2^k}\right)\right]$?
Some time ago I saw a problem somewhere (I cannot remember where) that goes something like this. Let $f:[0,+\infty)\to\mathbb{R}$ be a real function with $f(x)=f(\lfloor x\rfloor)$ for all $x\geq0$, $...
- 31
2
votes
1
answer
27
views
Area of a surface by means of a double integral
Considering the surface in $\mathbb{R^3}$ given by:
$x^2-y^2=1$; $x>0;-1<y<1;0\leq z \leq 1$
Calculate its area by a double integral via a parametrization of the surface.
Firsly I setted ...
- 67
1
vote
1
answer
77
views
Getting negative value of Stokes theorem integral
I was attempting the question:
Let $F(x,y,z) = (y, -x, 2z^2+x^2)$ and $S$ be the part of the sphere $x^2+y^2+z^2 = 25$ that lies below the plane $z = 4$. Evaluate the expression $\iint \operatorname{...
0
votes
0
answers
28
views
how to evaluate $\left[f(x) e^{-i \omega x}\right]_{-\infty}^{\infty}$ with $\lim \limits_{x \rightarrow \pm \infty} f(x)=0$
$\left[f(x) e^{-i \omega x}\right]_{-\infty}^{\infty}$ with $\lim \limits_{x \rightarrow \pm \infty} f(x)=0$.
$\left[f(x) e^{-i \omega x}\right]_{-\infty}^{\infty}=\left[f(x) (\cos (\omega x)+i \sin(\...
0
votes
0
answers
19
views
$g(\vec{x}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{a} \text{ where } 1 \le k \le m$ Is $g$ continuous at $\vec{a}$?
$$g(\vec{\mathbf{x}}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{\mathbf{a}} \text{ , where } 1 \le k \le m. \\\text{ Discuss the continuity of } g(\vec{\mathbf{a}}).$$
A ...
0
votes
1
answer
31
views
Disproving uniform continuity of a function using Cauchy continuity
From my understanding of uniformly continuous functions, they will, by definition, map Cauchy sequences to Cauchy sequences (thus preserving the Cauchy sequence in its transformation). If a function ...
- 355
1
vote
0
answers
39
views
Let $f\in AC[a,b]$, then the function $x\mapsto V_a^x(f)$ is absolutely continuous.
Let be $f\in AC[a,b]\subseteq BV[a,b]$ be an absolutely continuous function on $[a,b]$. We must prove that the function $$x\mapsto V_a^x(f)$$ is an absolutely continuous function.
We remember that $$...
- 812
-1
votes
0
answers
35
views
Arbitrary union of open sets proof in real analysis
Definition:
We say a set $U \subset \mathbb R $ is open if for every $x \in U $ there exists $ \epsilon > 0 $ such that $$(x-\epsilon , x+ \epsilon) \subset U$$
I proved that the union of two open ...
- 319
5
votes
1
answer
129
views
Showing that an upper sum is an upper bound of the improper integral
The question states to show that for $N = 1,2,3,...$ we have:
$$\sum_{k=1}^{N} \frac{1}{\sqrt{k^2+1}+k} \color{green}> \frac{1}{2}\ln{\frac{2N + 1}{3}}$$
My idea is to show the following
$$\sum_{k=...
- 386
0
votes
1
answer
41
views
For $x\in[0,1]$ find $\lim_{n\to\infty}\frac{n^2 x}{e^{n^2 x^2}}$
now this expression gives infinity over infinity so l'hospital rule can be used and the limit is zero which makes sense because the exponential is faster . but my question is what happens when x is ...
0
votes
1
answer
42
views
What does the notation $D$ mean here?
So what does the notation $D$ mean in the definition of relative entropy?
I try to explain the $D$ as $\nabla_{(\rho,u)}$, but it seems not right.
The article is "EXISTENCE AND UNIQUENESS OF ...
- 689
-3
votes
0
answers
54
views
Use sequential continuity to show that f = g everywhere [closed]
Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions, and that we have $f(x) = g(x) \ \forall \ x \in \mathbb{Q}$. Use sequential ...
0
votes
1
answer
53
views
A bounded function, which has a primitive function, that is not continuous in more that a finite number of points?
I am looking for an example of a function that is bounded on a compact interval and has a primitive function, but that is not continuous on that same interval in more than just a finite number of ...
- 21
0
votes
0
answers
24
views
A reference for the Weierstrass epsilon-delta definition of continuity
I am trying to locate a good reference related to the ${\varepsilon-\delta}$ definition of continuity. In particular, according to Wikipedia, the definition was stated respectively by Weierstrass and ...
- 241
1
vote
1
answer
30
views
Question regarding integration by substitution and Riemann sums
I was reading Advanced Calculus: A Geometric View and I liked how he explained in integration by substitution why the differential changes the way it does(both in push-forward and pullback ...
- 261
2
votes
0
answers
42
views
Stochastic process vs. random function
On the wiki page on stochastic processes, it says that they can be interpreted as random elements of a function space, which makes complete sense to me. However, it goes on to say that this ...
- 196
2
votes
0
answers
19
views
On the nonempty intersection of almost convex sets
Let $(X,\|\cdot\|)$ be a Banach space and $U_{X}$ its closed unit ball. According to Himmelber, Fixed points of compact multifunctions, a subset $B \subset X$ is said to be almost convex if given $\...
- 174
1
vote
1
answer
49
views
Gluing lemma for convex functions?
Let $A, B$ are two open intervals of real numbers such that $A\cap B\neq\emptyset.$ Suppose there are two convex functions $f: A\to\mathbb{R},$ and $g: B\to\mathbb{R}$ satisfying $f(x)=g(x)$ for all $...
- 17.9k
0
votes
0
answers
25
views
A doubt about the definition of absolute continuity on $\mathbb{R}$.
We say tha a collection of closed intervals in $\mathbb{R}$ is non-overlapping if their interiors are disjoint. A real- valued function $f$ on a finite closed interval $[a,b]$ is said to ve absolutely ...
- 812