Can a monotone differentiable function have uncountably many fixed points?...
I know that for a continuous function this is possible, but can we also construct monotone and differentiable, or even monotone and $C^{\infty}$ differentiable functions with uncountably many fixed...
View ArticleBijection From $D$ to $\overline{D}$ [closed]
Construct an explicit bijective map $ f: D \to \overline{D} $, where$$D := \left\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\right\}$$is the open unit disk in $ \mathbb{R}^2 $, and $ \overline{D}$ is...
View ArticleOn the continuity of the function $f(x)=\int_1^\infty \frac{\cos t}{x^2+t^2}dt$.
Let $F(x)=\int_1^\infty \frac{\cos t}{x^2+t^2}dt$. Then which of the following are correct?$f$ is bounded on $\mathbb R$$f$ is continuous on $\mathbb R$$f$ is not defined everywhere on $\mathbb R$$f$...
View Articlegive an example of a set that has exactly two accumulation points
I am new to this forum so please bear with me if I miss any information. I am currently taking advanced calculus using the textbook Introduction to Analysis (5th Ed) by Edward Gaughan. I am having a...
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Bump functions of maximal height under smoothness constraints
Suppose I want to find a function that is of maximal height at $0$ and of zero height outside of the unit ball. In other words, a bump function of maximal height.The constraint I put on this function...
View ArticleInfinite Product...
I've been looking at proofs ofEuler's Sine Expansion, that is$$\frac{\sin\left(x\right)}{x}=\prod_{k = 1}^{\infty}\left(1-\frac{x^{2}}{k^{2}\pi^{2}}\right)$$All the proofs seem to rely on Complex...
View ArticleWhat is the limit...
I need to compute for these special polynomials that are linear combinations of certain Jacobi Polynomials and are associated to some convergence acceleration methods of alternating...
View ArticleHow large can the set of turbulent points be?
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:$$\liminf_{r \to 0_+} \frac{|E \cap B_r...
View ArticleTranslation invariance of Lauricella integral
Given -$$f(z_0,\cdots,z_n)=\int_{\Gamma} \frac{d \zeta}{(z-z_0)^{\mu_0}\cdots(z-z_n)^{\mu_n}}$$where $\Gamma$ is an appropriate contour such that it has no poles on it. I need to show that...
View ArticleIf $A\subset \mathbb{N}$ is large, then does $\sum_{n\in A} \frac{\vert \sin...
If $A\subset \mathbb{N}$ is large, that is, $\displaystyle\sum_{n\in A} \frac{1}{n}$ diverges, then does $\displaystyle\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also?I know that...
View ArticleQuestion about the Inverse Function Theorem from Güler's *Foundations of...
I'm working through the Inverse Function Theorem as presented in Foundations of Optimization by O. Güler, and I came across a detail in the proof that I'm having trouble understanding. I’d really...
View ArticleIntro to Analysis, accumulation point proof [closed]
I am new to this forum so please bear with me if I miss any information. I am currently taking advanced calculus using the textbook Introduction to Analysis ($5$th Ed) by Edward Gaughan. I am having a...
View Articleconvex and differentiable imply twice differentiable
Let $f:\mathbb{R}\to\mathbb{R}$ be convex and differentiable, $a\in \mathbb{R}$ such that for all real $h$, we have : $f(a+h)=f(a)+hf'(a)+o(h^2)$.Show that $f''(a)$ exists and $f''(a)=0$.I have a proof...
View ArticleThe value of supinf when infsup is infinity
I have an optimization problem of the form:\begin{align*}\text{minimize } f(x) \text{ s.t. } x \in X \text{ and } g(x) \le G.\end{align*}I have $f$ as a continuous function, $X$ is both compact and...
View ArticleIs this a known space of functions?
Let $a>0$ and $f \in L^{p}([0,\infty[)$ be such that$$\sup_{x>0} x^{a}\, \| f \|_{L^{p}([0,x])}<\infty.$$It is easy to show that this is a norm. Is this a known function space or related...
View Article$\sum_{k=1}^{\infty} (-1)^k k$ is $(C,2)$ summable to $-\frac{1}{4}$
I am trying to show that $$\sum_{k=1}^{\infty} (-1)^k k = -\frac{1}{4}\quad (C,2)$$First I look at the partial sums:$S_1 = -1$, $S_2 = 1$, $S_3 = -2$, $S_4 = 2$...These will diverge to $\pm\infty$ so...
View ArticleHow to proof supremum of S belongs to S given distance between any two...
problem set questionHey, I've spent the better part of two hours on this problem and haven't found a good way to attack it. I understand it intuitively, I can't imagine a case where this isn't true,...
View ArticleQuestions about proof that every open subset $\mathcal{O}$ of $\mathbb{R}$...
I'm looking at the following proof in my textbook.Every open subset $\mathcal{O}$ of $\mathbb{R}$ can be written uniquely as a countable union of disjoint open intervals.Proof. For each $x \in...
View ArticleA property of Weak Solution
$u\in W_0^{1,2}(\Omega)$ is the weak solution to function $-\Delta u=F(\nabla u)$, where $F:\mathbb R^n\to \mathbb R$ is a map satisfying $|F(v)|\le |v|^2$.Prove: For each $\phi\in...
View Articlea real function $f(x)$ with $\zeta\in\mathbb{R}$ s.t....
The explicit problem is: if a real function $f(x)$ has the continuous third-order derivative over $\mathbb{R}$ then $\exists\,\zeta\in\mathbb{R}$ s.t. $f(\zeta)f'(\zeta)f''(\zeta)f'''(\zeta)\geq0$.If...
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