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80.129 real-analysis questions.

Question: If $f$ is differentiable at $x$, then for $\alpha\neq 1$, $f'(x)=\lim\limits_{h\to0}\frac{f(x+h)-f(x+\alpha h)}{h-\alpha h}$. My attempt: By applying the Secant Method, $f'(x)=\frac{f(x_n)-...

If we let $A_n = \{|X_n - X| \le \varepsilon\}$ for $\varepsilon > 0$ in Fatou's Lemmas for probability: $$P(\liminf A_n) \le \liminf P(A_n) \le \limsup P(A_n) \le P(\limsup A_n),$$ then I think ...

I recently came across an interesting conjecture that stated that the amount of positive integers $x$ on the interval $[1,n]$ such that $x$ does not have a $2$-middle divisor divided by $n$ approaches ...

Consider the Taylor series of a function $f(x)$ we usually assume $f(x)$ to be a polynomial: $$f(x)=a_0+a_1x+a_2x^2+a_3x^3...$$ But why not assume $$f(x)=a_0+a_{0.5}x^{0.5}+a_1x+a_{1.5}x^{1.5}+...$$ ...

My real analysis notes define a connected set $A \subset M$ of a metric space $M$ as a set where there does not exist open sets $U,V$ with 1) $ A \subset U \cup V$ 2) $ A \cap U \neq \emptyset$ 2) $ A ...

Let $d\in\mathbb N$, $\Lambda\subseteq\mathbb R^d$ be connected and open and $A:\Lambda\times\Lambda\to\mathbb R$ be twice continuously differentiable. Let $x,x'\in\Lambda$ and $h,h'>0$. How can we ...

Is the given following statement istrue /false ? Given $X = [-1,1] \times [-1,1]$ such that C ={$(x,y) \in X : xy= 0 $} and D ={$(x,y) \in X, x = {+}^{-} y$ }. Then $C$ is homeomorphic to $D$...

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with $\...

Today I was trying to do this exercise: Find for which $\lambda \in \mathbb{R}$ $$x+x^2=\arctan(\lambda x+x^2)$$ has exactly one solution. My attempt: Let's define $f(x)=x+x^2-\arctan(\lambda x + ...

Problem: Suppose $A \subset M$ is connected where $M$ is a metric space. Show that $A, \emptyset $ are the only subsets of $A$ that are clopen relative to $A$. My thoughts: The empty set is always ...

I'm struggling with a problem that seems very straightforward, so I apologize in advance if the answer is trivial. I'm trying to prove that: $$f(x) = \begin{cases} x+x^2 & x \in \mathbb{Q} \\ x-...

The (simplest form of the) Rademacher theorem reads as follows: Any Lipschitz continuous function $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue-almost everywhere differentiable. In other words: If ...

I'm done (a) and (b)(i) but I'm stuck on (b)(ii). I thought that since the dominant contribution is coming in at t=0 that I could approximate the sin in the logarithm and use a Taylor expansion, but ...

It can be shown that, for $t > 0$, the function \begin{equation} \mathbf{x}_p(t):= \begin{bmatrix} \dfrac{-1+4t-4t^2(\ln{t}+1)}{2t^2}\\ \dfrac{10t+8t^2(-\ln{t}-1)}{2t^2} \end{bmatrix} \end{equation}...

I have difficulty proving the following claim from a paper (a free version is here, see Lemma 2.4 on page 9): Let in a Banach space $X$ a sequence $\{x_n\}_{n=1}^\infty$ be given. Assume that for ...

In the paper "DYNAMICAL ASPECTS OF MEAN FIELD PLANE ROTATORS AND THE KURAMOTO MODEL" by L. Bertini, G. Giacomin, AND K. Pakdaman we read: $$r:=\Psi(2Kr),\quad\text{with}\quad\Psi(x):=\dfrac{\int_{\...

It's a well-known fact that $$\lim_{n\to\infty} (H_n-\log(n))=\gamma$$ Now, if I change things a bit and use the fact that $\displaystyle \Gamma \left( \displaystyle \frac{1}{ n}\right) \approx n$ ...

Here's the original question: Let $(a_n)$ be bounded. Assume that $a_{n+1} \ge a_{n} - 2^{-n}$. Show that $(a_n)$ is convergent. Okay, I know that if I can show that if the sequence is monotone, ...

Using the definition of the Hausdorff measure as $$H^\alpha(C) = \lim_{\delta\rightarrow\infty} H_\delta^\alpha(C) = \lim_{\delta\rightarrow\infty} \left( \inf\sum_{i=1}^\infty |\operatorname{diam}(...

Is there any example that a sequence in $l^p$ that converges weakly but not in a sense of norm for $1<p<\infty$?? There is a theorem that for $p=1$, if a sequence converges weakly, it converges ...

Problem : Show that the function $f(x) = \dfrac{2x^5-98}{(x-1)(x-9)}$ is uniformly continuous on $(2,8)$. I've tried doing this by a direct proof and using the definition to try and factor out a '$(x-...

If $X \subset \mathbb{R}^m$ have measure zero then, for all $Y \subset \mathbb{R}^n$, the cartesian product $X \times Y$ have measure zero in $\mathbb{R}^{m+n}$. Can anyone give a Hint, however that ...

I want to solve the following integral $$\int_0^{\infty}e^{-a(1+x^{m})^{\frac{2}{m}}}x\,{\rm d}x$$ where $a$ is real number but is not equal to $0$ and $m>2$. Any help or upper/lower bounds on ...

The functional sequence is given by; $$f_n(x)=ne^{-(x-n)^2}$$ So far all I have got is $$f(x)= \lim_{n\to \infty} ne^{-(x-n)^2} = 0 $$ i.e. it is pointwise convergent to zero. Question: Am I correct ...

I try to compute explicitly the Fourier transform of $f: z \mapsto 1/z^2 \in \mathbb{C} $ as a transform in $\mathbb{C}$, i.e. $$\mathcal{F}(f)(z)=\int_{\mathbb{C}}e^{-i2\pi\xi z}\frac{1}{\xi^2}d\xi$$...

Firstly I have these 2 conditions for arbitrary sequences {$x_n$}: Given any 1$\lt$q$\lt$p$\lt$ $\infty$, there is a real sequence {$x_n$} such that $\sum_{n=1}^{\infty}|{x_n}|^{p}$ is convergent ...

Give an example of a continuous function on $[a,b]$ not differentiable at only one point in $(a,b)$ not satisfying Lagrange Mean value theorem.

Let's say $G$ is some additive subgroup of $\mathbb{R}$ that has at least two elements. From what I understand, $G$ is then either dense in $\mathbb{R}$, or has some least positive element. What is ...

I wanted to find Example of uniform continuous unbounded function such $\lim_{x \to \infty}f(x) \neq \infty$ or $\lim_{x \to \infty}f(x) \neq -\infty$.But I did not get.I think there must be some ...

Q. For a continuously differentiable function $f:[a,b]\to \mathbb{R}$, define $$\|f\|_{C^1}=\|f\|_u+\|f^{\prime}\|_u.$$ Suppose $\{f_n\}$ is a sequence of continuously differentiable functions such ...

The functional series is given by;$$\sum_{j=0}^\infty \frac{\sin(jx)}{(2+x^2)^j} $$ I believe that for this question I should be using the Weierstrass M-Test. So far I have $$\frac{\vert \sin(jx)\vert}...

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 \int_0^...

What are your thoughts on this series? $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k)^2 \Gamma (n) }{\Gamma (2 k+n)}((\psi ^{(0)}(n)-\psi ^{(0)}(2 k+n)) (\psi ^{(0)}(k)-\psi ^{(0)}(2 ...

This question is a follow-up on this question. Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that at least one of $k,d$ is not even. Let $\Omega$ be an open subset ...

We consider the operator $$A=y\partial_x-\partial_xV(x)\partial_y+\frac{1}{2}(-\Delta_y+y^2)$$ where $V(x)$ is a polynomial defined in $\mathbb{R}^n$ Can someone please tell me $A$ is not elliptic?

If $f_n \to f$ is pointwise on $[a,b]$ and each $f_n$ is differentiable, and $(f'_n)$ converges uniformly on $[a,b]$ to some function $g$, I want to prove that $f'=g$. So if we let $\epsilon > 0 $ ...

I wanted to show uniform continuity of function $f(x)=\sqrt{x} \sin\sqrt{x}$ over $[0,\infty)$. I used all method know to me that continuity extension, definition, bounded derivative test, but not ...

I'm reviewing this epsilon delta proof for the continuity of the cubic root: but I can't see why is so evident that $\sqrt[3]{x^2}+\sqrt[3]{xc}+\sqrt[3]{c^2}\ge \sqrt[3]{c^2}$. Any help please? ...

Let $f,g: \mathbb{R}^n → \mathbb{R}^k$ be continuous functions and suppose that $D \subset \mathbb{R}^n$ is a dense set. If $f(x)=g(x)$ for every $x \in D$, then $f(x)=g(x)$ for every $x \in \mathbb{R}...

$\int\frac{2x+3}{x^2-4}dx=\int\frac{2x+3}{(x+2)(x-2)}dx$ So... $A=\frac{1}{4}$ $B=\frac{7}{4}$ $\frac{1}{4}\log_{10}|x+2|+\frac{7}{4}\log_{10}|x-2|+c$ The result should be: $\log_{10}|x^2-4|+\...

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f \colon \mathbb{R}_+ \times \Omega \to \mathbb{R}_+$ be a function such that: (i) For each $z \in \Omega$, the function $k \mapsto f(k,...

I am looking at a proof of the Lebesgue outer measure of the real line being countably subadditive. It states that for every set $E$ (whose outer measure is finite), there is a countable collection of ...

I've reviewed the literature but could not find a solution of the integral equation $$\int_{0}^{1} dx\frac{\phi(x)}{|x-y|^{\alpha}}+a\phi(y)=b.$$ for the function $\phi$ where $\alpha\in(0,1)$ and ...

Possible Duplicate: Subset of the preimage of a semicontinuous real function is Borel A real function $f$ on the line is upper semi-continuous at $x$, if for each $\epsilon > 0$, there exists $...

I would like to prove that for a LSC function, its epigraph is closed. I saw some longer proof here, but why would the following not hold ? : $f LSC := \liminf f(x_n) \ge f(x)$ when $x_n \rightarrow ...

Let $X$ be a metric space. A real-valued function $f : X \rightarrow \mathbb{R}$ is upper semicontinuous if it satisfies one of the followings: $(1)$ For all $c \in \mathbb{R}$, its preimage $f^{-1}(...

The following problem is stated as Exercise 22.A(iii) in the book Van Rooij, Schikhof: A Second Course on Real Functions. Let $f\colon [a,b]\to{\mathbb R}$ be Lebesgue integrable and upper ...

Determine the constants $a$ and $b$ in terms of the parameter $α$ in such a way that the multistep method $y_{n+1}=-ay_n - αy_{n-1} + hby'_n$ have the largest order possible. For what ranges of ...

Define the subset $S\subset\mathbb{R}$ to be equal to $\{a_1\mathbb{Z} + a_2\mathbb{Z} +\cdots+ a_n\mathbb{Z} \}$, where $a_1,a_2\cdots,a_n\in\mathbb{R}$ are rationally independent and $2\leq n<\...

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