Julia Wódka

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

27/10/15 17:13

Tuesday, November 10, 2015 17:15

\[\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 \(\mathbb{R}^\mathbb{R}\):

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.

*Room:*D1-215*Julia Wódka*

Speaker:Speaker:

*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 \(a < b\) and \(y\) is a number between \(f(a)\) and \(f(b)\), there exists an \(x_0\in(a, b)\) such that \(f(x_0) = y\)).
- 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 \({\rm diam} (f[J\cup \{x\}]) <\varepsilon\). - 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 \(f\)).
- 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.