Mean value properties for symmetrically differentiable functions
The authors define some modifications of Darboux property and mean value property, and examine connections between them for both symmetrically and approximately symmetrically differentiable functions. A function \(f:R\to R\) \((R\) --- the real line) is said to be strongly Darboux \((f\in D^ +)\) if \(f+g\) is Darboux for every linear function \(g(x)=cx\) \((c\in R)\). A function \(f\) is said to be weakly Darboux \((f\in D^ -)\) if for every \(a<b\), every \(C\) strictly between \(f(a)\) and \(f(b)\), and every \(\delta>0\) there is a \(c\in(a-\delta,b+\delta)\) such that \(f(c)=C\). It is shown, e.g., that for a symmetrically differentiable function \(f\in D^ +\) (an approximately continuous, approximately symmetrically differentiable function \(f)\) the following statements are equivalent: 1. \(f^ s\in D^ -\;(f^ s_{ap}\in D^ -)\); 2. for each \(a<b\), there exists \(\xi\in(a,b)\), such that \(f(b)-f(a)=f^ s(\xi)\) \((f(b)-f(a)=f^ s_{ap}(\xi)(b-a))\).
approximately symmetrically differentiable functions
scientific article; zbMATH DE number 68383