Mean value theorems for differences (Q942851)

The author offers the folowing mean value theorem. Let \(f: [a,b]\to\mathbb{R}\) be continuous (\(a<b,\;a,b\in\mathbb{R}\)). If there exists an integer \(n\geq 2\) such that (1)\(\;p=(b-a)/n\) then there exists a \(c\in[a-b,p]\) such that (2)\(\;(f(c+p)-f(c))/p=(f(b-a)/(b-a).\) If, however, \(p\) is not of the form (1) then there exists an \(f\in C^{\infty}\) for which (2) does not hold for any \(c\in[a,b-p].\) The rest of the paper contains results for the variants where \([a,b]\) is replaced by the circumference of a circle or by an \(m\)-dimensional interval \((m>1).\)