Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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1 vote
1 answer
523 views

Function space between uniform continuity and Hölder continuity

Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder ...
15 votes
2 answers
966 views

If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. We say $x \in \mathbb R^n$ is a strong Lebesgue point of $f$ if $$\lim_{r \to 0} \frac{\int_{B_r (x)} |f(y) - f(x)| \, dy}{r^{n+1+\...
4 votes
0 answers
68 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
0 votes
1 answer
183 views

Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]

Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
6 votes
2 answers
617 views

Forcing the uniqueness of a solution of an ODE

For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and $$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$ Let $y_n$ be the ...
22 votes
5 answers
2k views

Axiomatic construction of trigonometric functions

I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties: $\sin^2 x + \cos^2 x = 1$, $\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
2 votes
0 answers
107 views

What kind of points are left in the set with rationals subtracted, who contains all rationals and is null?

Let {$q_i$} be a list of all rationals, $U_{i,n}$ be an open interval centered at $q_i$ with length of $2^{-i}/n$. Then open set $\bigcup_{i}U_{i,n} $ has the length of $1/n$ and contains all ...
4 votes
1 answer
164 views

Estimate an improper integral

Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established? $$ \...
0 votes
0 answers
163 views

Lipschitz map on positive definite cone of $n$-by-$n$ matrices

A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
2 votes
0 answers
55 views

Existence of a suitable smooth kernel

Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
11 votes
1 answer
937 views

In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
13 votes
1 answer
2k views

Anti Arzela-Ascoli

Notation: We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence $f_n$ of real valued functions on $[0, 1] $ are equibounded if $\sup_{n \in \mathbb N}...
0 votes
1 answer
345 views

Approximation of Incomplete elliptic integral of first kind

How can we represent F(x,m) in the infinte polynominal of x,m? (Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia) More ...
6 votes
2 answers
740 views

If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous almost ...
1 vote
0 answers
43 views

Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem \begin{equation}\label{fainait ve} \dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
2 votes
1 answer
73 views

An inequality about the second-order difference

Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small \begin{...
1 vote
0 answers
92 views

Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
8 votes
1 answer
991 views

A seemingly trivial property of differentiable functions

NOTE. This is not really the question I wanted to ask. Somehow I forgot to mention that I am assuming $f$ is continuous. However, since Iosif's answer has been well-received I have left this question ...
7 votes
0 answers
475 views

A seemingly trivial property of continuous functions differentiable at the origin (PART 2)

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function such that $F(0)=0$, $F$ is differentiable at $0$ and $DF(0)$ is invertible. Is there an elementary way to show that for all $\epsilon>0$ ...
3 votes
1 answer
119 views

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$. Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
1 vote
2 answers
6k views

Continuous functions dense in $L_1$

If $X$ is a complete doubling metric space equipped with a complete probability measure $\mu$ such that all Borel sets are $\mu$-measurable, then $C_c(X)$ --- the continuous functions with compact ...
5 votes
2 answers
305 views

If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true ...
7 votes
1 answer
249 views

Does this "local time" type limit exist a.e. for $C^2$ functions?

For $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\...
0 votes
2 answers
488 views

A Jordan arc in the unit disk

Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
7 votes
2 answers
396 views

How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?

I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere. Here $\Omega$ is any ...
3 votes
1 answer
207 views

Continuity of Legendre transform

Let $I \subset \mathbb{R}$ be an interval, and $f_n, f: I \rightarrow \mathbb{R}$ a convex function; then its Legendre transform is the function $f^{\ast}: I^{\ast} \rightarrow \mathbb{R}$ defined by $...
3 votes
1 answer
212 views

Explicit bounds on the difference between Bernstein polynomials

Let $f:[0,1]\to [0,1]$ be continuous. Let— $$B_n(f)(x)=\sum_{k=0}^n f(k/n) {n \choose k} x^k (1-x)^{n-k},$$ be the Bernstein polynomial of $f$ of degree $n$. This question relates to the difference ...
4 votes
2 answers
439 views

About Euclidean distances

$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$. Let $d$ denote the Euclidean distance in $\R^n$. Do then ...
6 votes
0 answers
133 views

Injectiveness of a monotonic surjective mapping $\mathbb R^n \to \mathbb R^n$ with $\det J \neq 0$

Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\...
1 vote
2 answers
296 views

Closed formula for this sum $\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$ [closed]

How to calculate this sum $$\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$$ Thank you in advance
6 votes
2 answers
402 views

Lipschitz mappings, covering dimension

Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$? For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
0 votes
1 answer
241 views

Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
5 votes
2 answers
278 views

Does postcomposition with an absolutely continuous function preserve Lebesgue points?

Let $f: \mathbb R^n \to \mathbb R$ be a bounded measurable function, and $g: \mathbb R \to \mathbb R$ an absolutely continuous function. Question: Is it true that if $x \in \mathbb R^n$ is a Lebesgue ...
2 votes
0 answers
147 views

Fourier transform harmonic oscillator eigenstates

The normalized eigenfunctions of the quantum harmonic oscillator are $$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$ where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
2 votes
0 answers
67 views

Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
0 votes
1 answer
141 views

Realizability of metric matrices

We call an $n\times n$-matrix ${\bf A}\in \text{Mat}(n\times n, \mathbb{R})$ a metric matrix if ${\bf A}_{ii} = 0$ for all $i\in \{1,\ldots,n\}$, ${\bf A}_{ij} = {\bf A}_{ji}$ for all $i,j \in \{1,\...
4 votes
1 answer
326 views

Inequalities involving binary representation of integers

Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
2 votes
1 answer
291 views

Matching the integral of a function on smaller open sets

Let $f: [0, 1] \to \mathbb R$ be Lebesgue integrable with $\int_0^1 f \, d \mu = C.$ Question: For every $K$ with $0 < K \leq 1$, does there exist an open subset $U$ of $[0, 1]$ of Lebesgue measure ...
4 votes
1 answer
265 views

Equidistribution of distances of integer points to a circle

I have noticed in the following graph that the euclidean distance between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1:={}$Circle with radius 7 and shell with thickness 1) and the nearest point on the ...
5 votes
3 answers
501 views

How to prove this (corollary of) hyperplane separation theorem?

$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$. The theorem is as follows. If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...
1 vote
0 answers
98 views

Real analytic map with connected fibers

Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
5 votes
2 answers
1k views

Relationship between KL, chi-squared, and Hellinger

There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
3 votes
1 answer
196 views

Volume of 3-dimensional region

Let $G$ be bounded finitely connected domain in $\mathbb{R}^3$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature. Each sufficiently small open ...
2 votes
1 answer
145 views

Characterization of extendible distributions

I asked this question on Mathematics Stackexchange, but got no answer. I found the following question which characterize the extension of a distribution in $\mathbb{R}$: Let $f \in L_{\text{loc}}^{1}(...
6 votes
2 answers
3k views

Multivariable monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...
5 votes
1 answer
174 views

Intermediate value property for Sobolev functions

Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function. Question: For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\...
1 vote
0 answers
69 views

Control of solutions to nonlinear elliptic equations away from boundary

Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
1 vote
1 answer
175 views

Lipschitz aspect of a projection on the boundary of a convex

Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that \begin{...
3 votes
0 answers
134 views

A uniqueness result for the Neumann problem for the Laplace equation

Let $\Omega \subset \mathbb{R}^{3}$ be a $C^{1}$-domain, not necessarily bounded. Consider solutions $\phi : \overline{\Omega} \to \mathbb{R}$, $\phi \in C^{\infty}(\Omega) \cap C^{1}(\overline{\Omega}...
0 votes
1 answer
143 views

Determine if an integral expression is in $L^2(\mathbb{R})$

Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...

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