Articles of real analysis

Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$

This is a follow up to the great answer posted to Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral $$ I_{1}(x) = \int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$$ Fix a constant $c^* = r^{\frac{1}{2r+2}} $ and let $x^* = x^{\frac{1}{1+r}}$. […]

Increasing sequence of step functions converging to $\chi_G$ where $G$ is open in $$

Proposition 1. Let $I=[a,b]$ be a compact interval of $\mathbb{R}$. Let $G$ be an open set of $I$. Then there exists an increasing sequence of step functions $\{s_n\}$ defined on $I$ such that $s_n\nearrow \chi_{G}$ almost everywhere on $I$. I was trying to prove it, so I first considered the case of $G$ being connected […]

$\sup\{g(y):y\in Y\}\leq \inf\{f(x):x\in X\}$

Let $X$ and $Y$ be two nonempty sets and let $h:X\times Y\rightarrow \mathbb{R}$ have a bounded range in $\mathbb{R}$.Let $f:X\rightarrow \mathbb{R}$ and $g:Y\rightarrow \mathbb{R}$ defined by $$f(x)=\sup\{h(x,y):y\in Y\}$$ and $$g(y)=\inf\{h(x,y):x\in X\}$$Then can we prove that $$\sup\{g(y):y\in Y\} \leq \inf\{f(x):x\in X\}?$$

Uniform continuity question

Undoubtedly quite simple, but I’m stuck: If $f$: $[a, b)\longrightarrow\mathbb{R}$ is uniformly continuous, must $\lim\limits_{x\longrightarrow b}f(x)$ exist?

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times.

This question already has an answer here: Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times 7 answers

Defining the multidimensional Riemann Integral as a limit of certain sums

Currently, I am taking a course where we defined the multidimensional Riemann Integral of a map $f:\mathbb{R}^n \to \mathbb{R}$ as the limit $$\lim_{\varepsilon \to 0}\phantom{a}\varepsilon^n \sum_{x \in \mathbb{Z}^n}f(\varepsilon x)$$ If we require that $f$ is continuous and has a compact support, it is guaranteed that the above limit exists. Unfortunately, I do not know any […]

Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain?

Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain ? The motivation for this question is I have a sequence of functions $\{f_n\}$ where the number of maxima increases with $n$ and I am interested to know what happens to […]

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano’s Theorem, which, honestly, I find a bit of an overkill. For an informal proof, we could write something like this: “If $f$ is continuous on $[a,b]$ then […]

Taylor's series when x goes to infinity

Let $f(x) = \frac {x^3}{(x+1)^2}$. Find constants a, b, c, so that $f(x) = ax + b + \frac cx + o(\frac 1x)$ as $x$ goes to $\pm \infty$. So i know that i can’t take Taylor series as $x$ goes to infinity. So i am assuming i have to make some kind of substitution. […]

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different proof that only works when $X$ is sigma finite, but then it establishes also that the dual of $L^1(X)$ is […]