We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in , n ≥ 1, is a σ-limit set for some continuous map.
We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering of X such that is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.
For any with we provide a simple construction of an -Hölde function and a -Hölder function such that the integral fails to exist even in the Kurzweil-Stieltjes sense.
We evaluate the Fresnel integrals by using the Leibniz rule only on a finite interval.