Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?

I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those that do not have tangent anywhere but the example of curve that either touches itself or intersects itself at every … Read more

Sub-quadratic Kolmogorov-Arnold?

The Kolmogorov-Arnold representation theorem says, essentially, that when computing a continuous function, the only multivariate function you really need is addition. (Somewhat) more precisely, it says that for any continuous function f:Rn→R, we can write f as f(x)=f(x1,…,xn)=2n∑q=0Φq(n∑p=1ϕq,p(xp)) If we count function evaluations (the additions, the Φq‘s and the ϕq,p‘s), we perform O(n2) operations. Is … Read more

Points of differentiability of squared distance from a point in metric spaces

Here the link to the same question I posted on MSE with no answer. Let $(X,d)$ be a complete and separable metric space and let $I:=(0, + \infty)$. I recall the definition of absolutely continuous curve in this setting: we say that $u \in AC(I;X)$ if there exists $g \in L^1(I)$, $g \ge0$ a.e. s.t. … Read more

Induced maps on Hyperspace Topologies

If $X$ is a topological space let $2^X$ denote the set of closed subsets. There are multiple topologies one may equip $2^X$ with (in particular, I have in mind the Vietoris, Fell and similar topologies which agree with the topology induced by the Hausdorff metric when $X$ is compact), and I have been trying to … Read more

Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper1. A metric space X is called an Atsuji space if every real-valued continuous function on X is uniformly continuous. Strikingly I have found in the above paper that, X is an Atsuji space if and only if every bounded real-valued continuous function on X … Read more

Smoothness of the radius of convergence

Let (x↦an(x))n be a sequence of smooth functions defined on some fixed interval I. Consider the power series ∑n≥0an(x)tn and denote by R(x) its radius of convergence. Does there exist references in litterature dealing with continuity and smoothness of x↦R(x) ? Examples of questions I want to ask are as follows : Assuming 0<R(x)<∞ for … Read more

Find a continuous function with a prescribed continuity set

It’s known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book “Understanding Analysis” by Abbott is stated in page 128 that this property is “sharp”; that is, for every $D \subseteq \mathbb{R}$ in the class $F_{\sigma}$ there exists a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that … Read more

Density of the max set of a non-differentiable function

For f:[0;1]→R, let Mf:={x∈[0;1]∣f(x) is a local strict maximum of f}. It is easy to see that for any f, Mf is at most countable. It is also easy to see that there can be a continuous f such that Mf is infinite. The first question you can ask is the following: is there a … Read more

Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d’) and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)? Answer Yes, these are the so-called … Read more

Continuous non-constant function with infinite intersections with horizontal line on a compact interval?

The title might be misleading, but whether such a function exists is what boggles me about the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all $a<b $ satisfying $f(a)=f(b)$, there exists $c$ in $(a,b)$ such that $f(a)=f(c)=f(b)$. Prove that $f$ is monotonous on $\mathbb{R}$. What I’m intuiting about this problem … Read more