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Questions tagged [real-analysis]

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

4
votes
1answer
813 views

Is the absolutely continuous image of a nowhere dense set is also nowhere dense?

Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an absolutely continuous map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set? Remarks: The answer is "no" if $f$ is ...
33
votes
8answers
8k views

Can Cantor set be the zero set of a continuous function?

More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth? Some days ago I discovered that in this proof ...
9
votes
2answers
690 views

Partition of R into midpoint convex sets

We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$. My question is: is it possible to partition $\mathbb{R}$ ...
2
votes
0answers
335 views

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
14
votes
6answers
2k views

What's a natural candidate for an analytic function that interpolates the tower function?

I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
5
votes
0answers
535 views

continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
21
votes
4answers
9k views

About the Riemann integrability of composite functions

When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions. For the composite function $f \circ g$, He presented three cases: 1) ...
10
votes
2answers
2k views

Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
0
votes
1answer
357 views

a unique solution ? iteration involving conditional distributions

consider the following mappings, G and T, $y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$ $z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$ where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in ...
7
votes
1answer
1k views

Can a continuous, nowhere differentiable function have specified “shape” at every point?

I'm a bit embarrassed to admit that: a) This is a rather unmotivated question. b) I can't remember whether or not I've asked this before, but searching doesn't seem to turn anything up so ... ...
4
votes
6answers
3k views

existence of antiderivatives of nasty but elementary functions

In discussing with my honors calculus class the fact that some continuous elementary functions do not have an elementary antiderivative, I realized I was unsure whether every discontinuous elementary ...
2
votes
1answer
807 views

Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings, I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
88
votes
9answers
27k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
6
votes
8answers
5k views

Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
6
votes
4answers
3k views

completeness axiom for the real numbers

Do any treatises on real analysis take the following as the basic completeness axiom for the reals? "Let $A$ and $B$ be set of real numbers such that (a) every real number is either in $A$ or in $B$; ...
12
votes
1answer
2k views

Hausdorff Dimension and Hölder Continuity

Suppose we have a curve γ : [0,1] -> ℝn. It is well known that if this curve is Hölder continuous for some exponent α then the Hausdorff dimension of γ[0,1] is bounded above ...
12
votes
1answer
883 views

Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them. Background The Wiener integral is an analytic tool to define certain "...
4
votes
2answers
315 views

Embeddings of Weighted Banach Spaces

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
3
votes
1answer
324 views

“exchange” of real analyticity and integration

Sorry for the impreciseness of the title. It is merely meant for an analogy. Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For ...
2
votes
0answers
467 views

When deRham curve is bijection?

Motivation: Suppose we have deRham curve. From wikipedia: Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M: $d_0:\ M \...
9
votes
0answers
377 views

Evaluating Shintani cone zeta functions

Hi everyone I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ ...
6
votes
3answers
1k views

Dependence of error on mesh for Riemann sums

Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$, and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest $\delta > 0$ such that every Riemann sum arising from a ...
5
votes
1answer
569 views

Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set

For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
6
votes
4answers
539 views

Explicit bounds for the asymptotics of oscillatory integrals

Recall the following theorem (c.f. LC Evans, M Zworski, "Lectures on semiclassical analysis", Theorem 3.15, depending on the version): Theorem: Let $\varphi: \mathbb R^n \to \mathbb R$ be smooth and ...
8
votes
1answer
865 views

Why is this generality in Vitali's Lemma useful?

In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets? Vitali's Lemma: ...
26
votes
7answers
3k views

Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be. "Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
2
votes
1answer
1k views

How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?

I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
2
votes
2answers
292 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.
11
votes
2answers
686 views

Inequality in Gaussian space — possibly provable by rearrangement?

The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem. Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
3
votes
5answers
1k views

Cardinality of Equivalence Classes of Cauchy Sequences

What's the cardinality of a single equivalence class of Cauchy sequences in ℚ? To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
2
votes
3answers
873 views

How can I measure the Morse index in infinite dimensions?

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
2
votes
4answers
2k views

Splitting a space into positive and negative parts

Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
4
votes
2answers
589 views

Decomposition of Hölder continuous functions

Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is ...
20
votes
4answers
1k views

Which is the correct ring of functions for a topological space?

There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one. ...
38
votes
4answers
11k views

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in ...
3
votes
1answer
234 views

Asymptotically multiplicative functions and matrices

Hi, Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
64
votes
15answers
13k views

f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

The question is about the function f(x) so that f(f(x))=exp (x)-1. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://scottaaronson....
27
votes
25answers
41k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
62
votes
9answers
12k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
3
votes
1answer
1k views

Minkowski inequality

In the Wikipedia proof of the Minkowski inequality (http://en.wikipedia.org/wiki/Minkowski_inequality), the following inequality is used: |f+g|p ≤ 2p-1(|f|p+|g|p). I was just wondering if this ...
25
votes
5answers
5k views

Is there a natural measures on the space of measurable functions?

Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
5
votes
2answers
396 views

Characterizing the Radon transforms of log-concave functions

f:Rd→R≥0 is log-concave if log(f) is concave (and the domain of log(f) is convex). Theorem: For all σ on the sphere Sd-1 and r∈R, gσ(r) := ∫σ.x=rf(x)dS(x) is a log-...