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Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties? For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$. There exist $0< …

Apologies if this question is too basic for MO. I think it should be the case that for any decreasing $f \colon [A,\infty) \to [0,\infty)$ and $k \geq 0$, if $\int_A^\infty f(x) e^{kx} \, dx < \infty …

[This question is an extension of my question Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?. I'm asking i …

Anthony Quas has provided an example of a càdlàg function for which the jumps are not summable: As in https://math.stackexchange.com/questions/10257/, for any $S \subset (0,1]$ and $(x_t)_{t \in S} \i …

This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen el …

Let $V$ be a normed vector space, let $l \in V$, and let $(a_n)$ be a sequence in $V$. We say that $a_n$ is Cesàro-convergent to $l$ if $\frac{1}{n}\sum_{i=1}^n a_i \to l$ as $n\to\infty$. Now I will …

Let $P$ denote the following proposition: There exists a set $S$ of subsets of $\mathbb{R}$ such that $S$ is totally ordered by inclusion; each member of $S$ has no accumulation points; the union of …

I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of generalisation of expr …

Obviously in the case that $n=1$, we have $s(x,x)=0$ and so if $f'(x) \neq 0$ then $\frac{\partial h}{\partial y_1}$ doesn't exist at $(x,x)$. So I will assume that $n \geq 2$. Answer to Question 1. …

Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$. Fix $L \geq 1$. I will say that $\mu_n$ converges in op …