thoughts about $f(f(x))=e^x$
I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to...
View ArticleSelf Study - Folland Real Analysis Problem(2.1.3)
The following is a problem I solved recently from Folland's Real Analysis, in particular Problem(2.1.3). I have read a few solutions online, including on this site, but they are different from mine,...
View ArticleLet $n \geq 1$ and $P[x]$ in $E_n$ the space of monic polynomials of degree...
Let $n \geq 1$ and $P[x]$ in $E_n$ the space of monic polynomials of degree $n $. Show that $ \inf_{P \in E_n} \int_0^1 | P ( t ) | d t> 0 $.This exercise comes from excercise $30$ of this link and...
View ArticleWhy is $e^{x}$ not uniformly continuous on $\mathbb{R}$?
It seems intuitively very clear that $e^{x}$ is not uniformly continuous on $\mathbb{R}$. I'm looking to 'prove' it using $\epsilon$-$\delta$ analysis though. I reason as follows:Suppose $\epsilon >...
View ArticleBennett's Inequality to Bernstein's Inequality
Bennett's Inequality is stated with a rather unintuitive function,$$h(u) = (1+u) \log(1+u) - u$$See here. I have seen in multiple places that Bernstein's Inequality, while slightly weaker, can be...
View ArticleOddity in definition(s) of quasi compact operator
I was wondering about the general definition of a quasicompact operator. There seem to be two main ones floating around in the literature, and I am not sure they are equivalent. The first, and...
View ArticleGiven $ T \in \mathbb{R}$,Find all differentiable functions that satisfy...
Given T $\in \mathbb{R}$, Find all differentiable functions that satisfy $$f'(x) = f(x + T)$$My Attempt. Firstly, I tried the exponential function $y=e^{ax}, T=\frac{\ln a}{a} \in (-\infty, 1/e]$...
View ArticleThe set of all rational points in the plane is a countable set
From Kolmogorov's Introductory Real Analysis. I am doing some self-study and would like some feedback on whether my proof is correct.I am using that the set of rational numbers is countable as given,...
View ArticleHow to sum $ \displaystyle\sum_{n=1}^\infty \sum_{\substack{j=1...
As the title suggests, I am trying to work out a double sum where one of the indexes has a condition of coprimality, meaning the index doesn't sum whenever $\mathrm{gcd}(2n,j) \ne 1$. The sum is the...
View ArticleProduct of a concave function and a decreasing function
Let $f(x)$ and $g(x)$ be two positive continuous functions.Function $f(x)$ is concave at $x_1$ and function $g(x)$ is $<1$ and is decreasing; $g(0)=1$ and $f(0)=0$.Define function $h$ as the product...
View ArticlePartitioning ℝ into sets $A$ and $B$, such that the measures of $A$ and $B$...
Motivation: I want to partition $\mathbb{R}$ into sets $A$ and $B$, where the measures of $A$ and $B$ in each non-empty open interval have positive ratios with an upper and lower bound, such that the...
View ArticleEvaluate limits of multivariable function going to infinity [closed]
Please show me how to evaluate limit of this function$\lim\limits_{(x,y)\to(+\infty,+\infty)}\frac{x+y}{x^2-xy+y^2}$.I think it doesn't exist but I don't know how to solve it.
View ArticleSumming inverse squares of primitive Pythagorean triples
Part of the solution was already answered here:Sum of the inverse squares of the hypotenuse of Pythagorean trianglesAs we all know, a primitive Pythagorean triple is determined by...
View ArticleExample of little $\alpha$ Hölder function that is not $\beta$ Hölder, for...
What functions are in the set $$ c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$A single example will do, but the more the merrier.This question was natural to me after writing this...
View ArticleOpen and connected subsets of $\mathbb{R}^n$ are arcwise connected
The result is well-known in a general setting (for path-connected Hausdorff topological spaces), but the proof is no picnic.I can imagine that for a connected and open subset of $\mathbb{R}^n$ the...
View ArticleProof of continuity from above using continuity from below
I'm reading a proof of continuity from above using continuity from below. Let $(X,\mathscr{A},\mu)$ be a measure space. Suppose $A_{1} \supseteq A_{2} \supseteq \cdots $ is a sequence of decreasing...
View ArticleInfinite polynomial interpolation for $f(x) ={}^x2$
As we all know, there is a thing about polynomial interpolation when using equally-spaced nodes on certain functions called Runge's phenomenon. However, there are a lot of functions that do not have...
View ArticleExample of a premeasure that is not a measure
I'm reading an example of a premeasure that is not a measure. We define an algebra on $X = \mathbb{N}$ as the collection of all finite and co-finite sets. We let $\mu$ be a premeasure on this algebra...
View ArticleSubset of separable metric space can have at most a countable amount of...
Let $(X,d)$ be a separable metric space. Prove that every subset $Y \subset X$ can have at most a countable amount of isolated points.Attempt at proof: Let $Y$ be an arbitrary (non-empty) subset of...
View ArticleWhy it is important to know the projection onto epigraph of a function?
Why, in general, someone should be interested in finding projections onto epigraph(f), where $f:X\to\mathbb{R}$ is a given function? ($X$ Hilbert space).I heard about that in problems related to convex...
View Article