# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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### What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
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### Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
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### Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
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### A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
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### Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$\sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
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### Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
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### On an asymptotic improvement of AMM problem 11145 (April 2005)

Motivation Motivated by this question, I tried improve the inequality $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le2\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ asymptotically. In other words, with ...
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### proof verification: $f(x) = 1/x$ is not uniformly continuous on the open interval (0,1).

I've written a proof that $f\left(x\right)=\frac{1}{x}$ is not uniformly continuous on the interval $(0,1)$ and would like to know if it is correct. Here's what I've got. In order to show a ...
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### Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt$ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
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### Understanding the Legendre transform

In physics, I've seen the Legendre transform motivated by "changing the variable $x$ of a function $x \mapsto f(x)$ to the variable $u = \frac{df}{dx}$." I don't quite see what that means and why the ...
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### How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}u\,dy.$$ But for the mean-value property of the Heat equation, ...