On equi-derivatives (Q1374540)

A family of locally Henstock-Kurzweil integrable functions \(f_s:\mathbb R \to \mathbb R\), \(s\in S\) is called a family of equi-derivatives at \(x\in \mathbb R\) if for every \(\eta >0\) there is an \(r >0\) such that for every \(h\), \(0<|h|<r\) and \(s\in S\) we have \(|\frac {1}{h}\int_x^{x+h}f_s(t)dt - f_s(x)|< \eta\) and the family is approximately equicontinuous if for every \(\eta >0\) there is a set \(B\in \mathcal T_d\) such that \(x\in B\) and for all \(t\in B\) and \(s\in S\) the inequality \(|f_s(t) - f_s(x)|<\eta\) holds. (\(\mathcal T_d\) is the so called density topology given by measurable sets for which every \(x\in A\) is a density point of \(A\).) Those two concepts are compared in the paper as well as some properties of functions defined on \(\mathbb R^2\) are studied.