K101 K101 FSv ČVUT

Seminář katedry matematiky

Aktuálně Přednášky Fotogalerie Program

Seminář katedry matematiky

Seminář katedry matematiky


Letní semestr 2017/18

Zde najdete stručný popis již proběhnuvších přednášek.

15. 5. 2018
B. Hanson

Název přednášky: An Introduction to the Little Lip Function

Přednášející: Bruce Hanson

Department of Mathematics, Statistics, and Computer Science, St. Olaf College, USA 

Abstrakt: Given a continuous function $f:\rr \to \rr$ with $\ds M_f(x,r)=\frac{\sup_{|x-y|\le r}|f(x)-f(y)|}r$, the so-called "Big Lip" and "Little Lip" functions are defined as follows: $$\ds\Lip f(x)=\limsup_{r\to 0^+}M_f(x,r) \quad\quad \ds\lip f(x)=\liminf_{r\to 0^+}M_f(x,r)$$ The Rademacher-Stepanov Theorem tells us that $f$ is differentiable almost everywhere on $L_f=\{x \,|\,\Lip f(x)<\infty\}$. On the other hand, as Balogh and Csörnyei showed, this theorem no longer holds if we replace $L_f$ with $l_f=\{x \,|\,\lip f(x)<\infty\}$. In this talk, I consider the problems of characterizing the sets $E\subset \rr$ for which there exist a continuous function $f$ such that $l_f=E$ as well as characterizing the sets of non-differentiability for functions $f$ with $l_f=\rr$. I will also examine some additional questions about the relationship between $L_f$ and $l_f$.

PDF version is located here.

Design stránek: Stanislav Olivík, 2013