Julia Wódka

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

Tuesday, November 10, 2015 17:15

Room: D1-215

Speaker:
Julia Wódka

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 $$\mathbb{R}^\mathbb{R}$$:

1. Darboux functions ($$f\in\mathscr{D}$$ if whenever $$a < b$$ and $$y$$ is a number between $$f(a)$$ and $$f(b)$$, there exists an $$x_0\in(a, b)$$ such that $$f(x_0) = y$$).

2. quasi-continuous functions ($$f\in\mathscr{Q}$$ if it is quasi-continuous at any point $$x\in\mathbb{R}$$).

Function $$f$$ is quasi-continuous at $$x \in \mathbb{R}$$ if for any open interval $$I\ni x$$ and each $$\varepsilon>0$$ there exists a nontrivial interval $$J\subset I$$ such that $${\rm diam} (f[J\cup \{x\}]) <\varepsilon$$.

3.   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 $$f$$).

4. 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 $$f(x)=y$$).

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
$p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.$
$p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.$
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.