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3 votes
1 answer
221 views

Closed subset of unit ball with peculiar connected components

Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball. Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii? i) $\{0\}$...
user_1789's user avatar
  • 722
2 votes
0 answers
94 views

Real analytic periodic function whose critical points are fully denegerated

I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
Jianxing's user avatar
9 votes
2 answers
670 views

Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?

The question is as in the title: Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
Isaac's user avatar
  • 2,331
1 vote
1 answer
216 views

Examples of $C^{k,1}$ functions which are not $C^{k+1}$?

I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which: $$ \|...
ABIM's user avatar
  • 5,001
0 votes
1 answer
306 views

Uniform approximation of indicator function of a point

Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that $g_n$ converge uniformly ...
Bernard_Karkanidis's user avatar
4 votes
2 answers
522 views

Is there a Borel-measurable function which maps every interval onto $\mathbb R$?

Using AC, one easily defines a function $F:\mathbb R\to \mathbb R$ such that the $F$-image of any real interval $(a,b)$ ($a<b$) is equal to $\mathbb R$. (Equivalently, the $F$-preimage of any real ...
Vladimir Kanovei's user avatar
3 votes
1 answer
208 views

What is a non-trivial example of an unbounded subdifferential?

Let $f: X \to [ -\infty, \infty]$ be some function, Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$, $$\partial f(x)$$ is "unbounded"? (trivial examples ...
Sin Nombre's user avatar
1 vote
1 answer
143 views

Original examples of functions of slow increase in the spirit of Jakimczuk

I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(...
user142929's user avatar
2 votes
1 answer
138 views

Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements

Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$ for ...
user142929's user avatar
1 vote
1 answer
276 views

Examples of Steffensen's inequality at undergraduated level studies

I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
user142929's user avatar
0 votes
0 answers
68 views

Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question. We consider the ...
user142929's user avatar
3 votes
2 answers
198 views

Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
Y.B.'s user avatar
  • 391
-1 votes
1 answer
148 views

How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$ Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
Math Learner 's user avatar
2 votes
1 answer
84 views

How to choose function $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$?

Can we expect to choose a function $f:\mathbb R \to \mathbb R$ (nonzero compactly supported) so that $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$ for all $x\in \mathbb R$ and $n\in \mathbb Z$?...
Math Learner 's user avatar
2 votes
3 answers
580 views

For every monotonically-decreasing non-negative function $ f $, does there exist a function $ g $ so that $ f g $ is integrable? [closed]

Let $ f $ be a monotonically-decreasing non-negative function satisfying $ \displaystyle \lim_{x \to \infty} f(x) = 0 $. Is it true that the following claim holds? Claim: There exists a function $ ...
Spinorbundle's user avatar
  • 1,889
15 votes
3 answers
1k views

Can integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function $$ g:(a,b)\to\mathbb{R},\qquad ...
H. Berbeleque's user avatar
18 votes
6 answers
3k views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
user23078's user avatar
  • 1,594
2 votes
2 answers
954 views

Function with all but mixed second partial derivatives twice differentiable?

Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
Tiffy's user avatar
  • 97
9 votes
1 answer
738 views

Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
Syang Chen's user avatar
10 votes
1 answer
764 views

Nondifferentiability set of an arbitrary real function

A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...
LostInMath's user avatar