80.129 real-analysis questions.

How is the Taylor formula applied here?

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 is true /false relating to homeomorphic?

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 $D$...

1 Relative Topology and clopen sets

1 answers, 31 views real-analysis general-topology
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 ...

14 Computing a limit involving Gammaharmonic series

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 $\displaystyle \Gamma \left( \displaystyle \frac{1}{ n}\right) \approx n$ ...

1 Showing that a sequence is monotone and thus convergent

1 answers, 18 views real-analysis sequences-and-series
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, ...

Is a Sobolev map with smooth minors smooth on the whole domain?

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 ...

Elliptic operator [on hold]

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?

3 Uniform convergence and differentiable functions proof

1 answers, 762 views real-analysis uniform-convergence
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 $\epsilon > 0$ ...

1 Uniform Continuity of function $f(x)=\sqrt x\sin\sqrt x$ on $[0,\infty)$

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 ...

4 Cubic root epsilon delta proof

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? ...

-3 Dense set and continuous functions [on hold]

1 answers, 18 views real-analysis continuity

1 “Upper derivative” of indefinite integral of upper semicontinuous function

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 ...

1 Determine the constants $a$ and $b$ in terms of the parameter $Î±$ in such a way that the multistep method have the largest order possible

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

2 Is the span of rationally independent real numbers dense in $\mathbb{R}$

1 answers, 270 views real-analysis functional-analysis
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<\...