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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3
votes
0answers
174 views

Example showing that area is discontinuous in the 2-variation seminorm

The $p$-variation seminorm (where $p \ge 1$) of a continuous curve $\alpha: [0,1] \to \mathbb{R}^2$ is defined as the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_n = 1$ of: $\left(\...
15
votes
3answers
7k views

the dual space of C(X) (X is noncompact metric space)

It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f: X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
0
votes
0answers
245 views

Jacobian of the inversion map

Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
5
votes
1answer
529 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
4
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0answers
105 views

rank of a C^1 map

I saw this three star problem in Hirsch .. If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank ...
5
votes
2answers
516 views

Asymptotic behaviour of $\int f(t)^a\cos(at)dt$

Are there any known necessary or sufficient conditions such that $$\lim_{a\rightarrow \infty}\int_{-1}^1f(t)^a\cos(at)dt=0$$ where $f:[-1,1]\rightarrow[1,\infty)$ is an even smooth concave real ...
24
votes
4answers
2k views

“Converse” of Taylor's theorem

Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for ...
2
votes
1answer
286 views

A raceway problem

Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set $S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway" My question is finding the shortest path in $S$...
0
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0answers
652 views

Sigma algebra generated

Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma ...
3
votes
1answer
837 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that $$\sum_{i=1}^...
10
votes
2answers
914 views

Does Rolle's Theorem imply Dedekind completeness?

I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
3
votes
2answers
2k views

Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs, Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
9
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0answers
292 views

Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
3
votes
3answers
524 views

Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ?

This question was originally asked in stackoverflow (https://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial) but as it has ...
1
vote
1answer
327 views

Property Sigma Algebra [closed]

Is the set { $ \cup_{i \in \mathbb{N}} C_{i} \times D_{i} : C_{i} \in \mathcal{L} \ , D_{i} \in \mathcal{B}^{n} \ $ } a sigma algebra on $\mathbb{R} \times \mathbb{R}^{n}$ ?
7
votes
2answers
2k views

Absolute continuity on $R^{n}$

I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$. I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
10
votes
2answers
2k views

Multi-dimensional moment problem

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
0
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1answer
963 views

surjective function from non-measurable sets

let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval ...
16
votes
3answers
3k views

Which functions have all derivatives everywhere positive?

Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$. The only examples I can ...
1
vote
3answers
557 views

Does complete monotonicity of f imply log-concavity of f?

Let f be a completely monotonic function with $f(0)=1$, that is, $ f:[0, \infty) \rightarrow (0,1] $. My question is: Is f log concave, that is, is $(logf)''<0$ or equivalently $ f f''< f'^2 $....
1
vote
2answers
941 views

Is there a periodic function without minimum period such that all the possible periods are irrationals? [closed]

Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\...
11
votes
5answers
2k views

analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
11
votes
2answers
1k views

Implicit function theorem at a singular point?

Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...
0
votes
1answer
534 views

Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]

Is there a difference between $L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ? Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue $\sigma$-algebra $\...
4
votes
2answers
1k views

Stone-Weierstrass for monotone functions

Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$. Does it follow that that there exists a real ...
0
votes
1answer
330 views

Under what condition will this set contain a limit point of [0,1)?

Let $T_1,T_2,....T_n$ be numbers such that $T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
0
votes
1answer
299 views

Modulo dynamics on [0,1)

For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\...
2
votes
2answers
3k views

Bounded sequences with divergent Cesàro mean

It is well known that there are bounded sequences with divergent Cesàro mean, i.e., a bounded $a_n$ for which $$\lim_{N\to\infty} c_N = \frac{1}{N}\sum_{n=1}^N a_n$$ does not exist. A simple example ...
1
vote
1answer
1k views

On an eigenvalue inequality

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| \...
5
votes
3answers
4k views

Zeros of “exponential” function

Define ${f}_{i}(x) = \sum_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$. For integer $x \geq i$, ${f}_{i}(x)$ reduces to ${f}_{i}(x)=i!S(x,i)$, where $S(x,i)$ is ...
5
votes
2answers
553 views

On the uncountability of zero sets

If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain. I ...
1
vote
2answers
391 views

A smoothness of $f(\sqrt[p] x)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function let $p \in \mathbb{N}$, $p \geq 2$. Assume that $f^{(k)}(0)=0$ for all $k \notin p \mathbb{N}$. Is it true that then $g(x)=f(\sqrt[p] x)$...
3
votes
1answer
414 views

Hausdorff measure on product spaces of p-adic integers

This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
2
votes
0answers
6k views

Dual space of continuous functions

Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
4
votes
2answers
3k views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
2
votes
1answer
251 views

Quotients of perfect powers separated by an integer

Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
5
votes
2answers
1k views

non-maximal prime ideal in the ring of continuous functions

Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\...
4
votes
2answers
345 views

Heights of several interesting posets

Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$). Define several sets of total functions, in each ...
4
votes
1answer
610 views

Does such a smooth function exist?

I am looking for a $C^\infty $ function $g:\mathbb{R}^3\to \mathbb{R}^3$ such that $g(x)=0$ for $|x|\le 1$ and $g(x)=x$ for $|x|\ge 2$. Certainly such $g$ can be constructed, but I also want it to ...
8
votes
2answers
916 views

Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
2
votes
1answer
430 views

Showing the derivative of this function is equal to $0$ a.e [closed]

Define $f:[0,1]\to [0,1]$ by $f(0)=0$, and $$f(x)=\sum\limits_{r_n\le x} 2^{ -n }$$ with $0\lt x\le 1$ where $[r_n]_{n\in \mathbb{Z^+} } = \mathbb{ Q} \cap (0,1) $. How to show that the derivative $...
21
votes
3answers
3k views

Prime ideals in the ring of germs of continuous functions

We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point. Can anyone think of a single ...
3
votes
2answers
2k views

mean value theorem for operators

This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \...
1
vote
2answers
2k views

Continuation of a smooth function

Setting Suppose I have two bounded open domains $\Omega' \subset \Omega \subset \mathbb{R}^n$ (I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are $C^\...
2
votes
0answers
840 views

Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
2
votes
0answers
472 views

Characterization of weak Lebesgue spaces [closed]

I would be interested to know whether the following is true: Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a ...
10
votes
4answers
4k views

The metric space associated to a measure space

Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \...
8
votes
2answers
737 views

Patching together homeomorphisms: how badly can it fail?

Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...
19
votes
2answers
1k views

Constants for Rolle's Theorem applied to polynomials

Rolle's Theorem states that $f(1/2)=f(-1/2)+f'(x)$ has a root in the open real interval $(-1/2,1/2)$ if $f$ is continuous and differentiable. How large can the absolute value of such a root $\xi$ be ...
0
votes
1answer
206 views

Special functions on the unit disk

Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk. We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following: 1) it is a ...

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