Julia Wódka: Comparison of some families of real functions in sense of porosity

Thursday, January 1, 1970, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Julia Wódka (Łódź University of Technology)

Title: Comparison of some families of real functions in sense of porosity

Abstract:

We consider set $\mathbb{R}^\mathbb{R}$ with uniform convergence metric, i.e:
\[\rho(f,g)=\min{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|}\quad \text{for $f,g\in\mathbb{R}^\mathbb{R}$}\]
and the following subsets of $\mathbb{R}^\mathbb{R}$:

Darboux functions ($f\in\mathscr{D}$ if whenever $a <b$ there exists a nontrivial interval $J\subset I$ such that ${\rm diam} (f[J\cup \{x\}]) <\varepsilon$.

Świątkowski functions ($f\in\mathscr{\acute S}$ if for all $a<b$ with $f(a)\ne f(b)$, there is a $y$ between $f(a)$ and $f(b)$ and an $x\in (a,b) \cap \mathcal{C}(f)$ such that $f(x)=y$, where $\mathcal{C}(f)$ denotes the set of all continuity points of function $f$).

strongly Świątkowski functions ($f\in\mathscr{\acute S}_s$ if for all $a<b$ and each $y$ between $f(a)$ and $f(b)$ there is an $x\in (a,b) \cap \mathcal{C}(f)$ such that $f(x)=y$).
\end{itemize}

The aim of this is to compare this sets in terms of porosity.

Let $(X,d)$ be a metric space, $x\in A\subset M$, and $r\in\mathbb{R}_+$. We define
\[\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}\]
and \[p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
\[\Bigl(p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\Bigr)\]
Quantity $p^u(M,x)$ is called upper porosity of $M$ at the point $x$.
We say that $M$ is upper $p-$porous if $p=\inf\{p^u(M,x):\ x\in M\}>0$.

Analogously we define lower porosity.

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