Monotonicity theorems for generalized Riemann derivatives (Q584433)

Let \(a_ i,b_ i\) be real numbers \((i=1,...,n)\), \(\sum^{n}_{1}a_ i=0\), \(b_ 1<b_ 2<...<b_ n.\) For a continuous function \(f:[a,b]\to {\mathbb{R}},\) we write \(D^ 1f(x)=\lim_{h\to 0}(\sum^{n}_{1}a_ if(x+b_ ih))/h\) for \(x\in (a,b);\) if lim is replaced by \(\overline{\lim}\) or \underbar{lim}, we write \(\bar D^ 1\) or Ḏ\({}^ 1\), respectively, and the symbols \(D^ 1_+,\bar D^ 1_+,\underline D^ 1_+\) are used if \(h\to 0\) is replaced by \(h\to 0+.\) All these values depend on the choice of n and \(a_ i,b_ i.\) Main results: If \(\underline D^ 1_+f(x)\geq 0\) and \(f'_+(x)>-\infty\) for every \(x\in (a,b)\) then f is increasing. For \(n=3,\) the authors give necessay and sufficient conditions for \(a_ i,b_ i\) in order that \(\underline D^ 1_+f(x)\geq 0(x\in (a,b))\) should imply that f is increasing. They present partial results for the question whether\(\underline D^ 1f(x)\geq 0(x\in (a,b))\) implies or not that f is increasing.