The symmetric derivative and the Darboux property
The symmetric derivative \(f^ s\) of the functin \(f: R\to R\) (R - the real line) is defined by \(f^ s(x)=\lim_{h\to 0}(f(x+h)-f(x-h))/2h.\) The function f is said to belong to class MVT (of functions fulfilling the mean value theorem) if for any \(a,b\in R,\quad a<b,\) there exists \(z\in [a,b],\) such that \(f(b)-f(a)=f^ s(z)(b-a),\) and, whenever for some \(x\in R,\quad \lim_{t\to x+}f^ s(t)\quad (\lim_{t\to x-}f^ s(t))\) exists, it equals \(f^ s(x)\). Let h be a locally bounded symmetric derivative. Then h has the Darboux property if and only if there exists a function \(\mu\in MVT\) with \(\mu^ s=h\). This statement gives a partial answer to a problem raised by \textit{L. Larson} [Real Anal. Exch. 9, 154-178, Query 172 (1984; Zbl 0574.26005)].
scientific article; zbMATH DE number 4038126