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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5
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1answer
1k views

Hanner's inequalities: the intuition behind them

Hanner's inequalities in the theory of $L^p$ spaces (see http://en.wikipedia.org/wiki/Hanner's_inequalities) look hard to come-up with at the first glance. Their proof (say, the one in Lieb & Loss ...
6
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1answer
730 views

Inverse function theorem for DC-functions

I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it... Definitions: Let $h:\mathbb R^n\to\mathbb R$ ...
-3
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1answer
464 views

A problem regarding definition of p-norm [closed]

Let ${\bf x}=(x_1,...,x_n)$, the p-norm of x is $(|x_1|^p+...+|x_n|^p)^{1/p}$. If one of the components of x is 0, there will be exponential of the form $0^p$. If p is an irrational, $x^p$ is only ...
13
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7answers
27k views

Real analysis has no applications?

I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
17
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12answers
3k views

Looking for an interesting problem/riddle involving triple integrals.

Does anyone know some good problem in real analysis, the solution of which involves triple integrals, and which is suitable for second semester Analysis students? Thanks!
3
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2answers
633 views

Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
2
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2answers
1k views

Examples of deterministic processes of quadratic variation which are of unbounded variation

In [Föllmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process ...
5
votes
1answer
330 views

The set of non-smooth points of a convex function is (m - 1)-rectifiable

I am looking for a reference to the following result. Let $f:\mathbb R^m\to\mathbb R$ be a convex function. Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-...
73
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8answers
10k views

Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
4
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4answers
4k views

The characteristic (indicator) function of a set is not in the Sobolev space H1

Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$ And if so, what is the best/...
7
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3answers
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Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...
4
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1answer
843 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 ...
35
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8answers
9k 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
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2answers
696 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
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0answers
340 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
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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
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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 ...
24
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4answers
10k 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
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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
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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
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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
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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
813 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 ...
90
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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
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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
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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
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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 ...
13
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1answer
899 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
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2answers
320 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
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1answer
326 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
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0answers
476 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
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0answers
382 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
575 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
546 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
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1answer
868 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
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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
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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
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2answers
293 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.
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2answers
693 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
876 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
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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
601 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
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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
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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
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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....
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26answers
42k 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
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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?