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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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
609 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 ...
21
votes
4answers
2k 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
12k 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
235 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)...
65
votes
15answers
14k 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....
29
votes
26answers
45k 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
2k views

Minkowski inequality

In the Wikipedia proof of the Minkowski inequality (http://en.wikipedia.org/wiki/Minkowski_inequality), the following inequality is used: $$|f+g|^p\leq2^{p-1}(|f|^p+|g|^p).$$ I was just wondering if ...
29
votes
5answers
6k 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
403 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-...

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