Positive derivatives and increasing functions (Q913966)

The author presents a simple proof for the following statement: if \(f\) is continuous in \([a,b]\) and \(Df>0\) on \((a,b)-S\), where \(S\) is countable, then \(f(b)>f(a)\); here \(Df\) denotes either the right upper Dini derivative or the symmetric derivative. The method of the proof is classic [and furnishes stronger results, see \textit{S. Saks}: Theory of the integral (1937; Zbl 0017.30004), Theorem 7.1].