Let E be a path system, i.e. \(E:R\to 2^ R\) and \(x\in E_ x\) and x is a limit point of \(E_ x-\{x\}\) for each \(x\in R\). Let \(h=\{h_ n\}^{\infty}_{n=1}\) be a sequence of nonzero numbers and \(\lim_{n\to \infty}h_ n=0\) and let \(E_ x=\{x+h_ n:n\in N\}\cup \{x\}\) for each \(x\in R\). Then a real function f of a real variable has at u the path derivative \(f_ E'(u)\) iff the limit \[ f_ E'(u)=\lim_{y\to u,y\in E_ u}(f(y)-f(u))/(y-u) \] exists and by the sequential derivative \(f_ h'\) we mean \[ f_ h'(u)=\lim_{n\to +\infty}[f(u+h_ n)-f(u)]/h_ n. \] In connection with some results of A. Khintchine and G. H. Sindalovskij which are not true, the authors discussed the following two natural questions: a) What information about \(f_ E'\) and \(f_ h'\) on a set A implies that f is differentiable or approximately differentiable a.e. in A ? b) When such derivatives exist on a set A, on which the approximate derivative also exists, what conditions will ensure that \(f'_{ap}(x)=f_ E'(x)\) or \(f'_{ap}(x)=f_ h'(x)\) a.e. in A ?
In this paper there are proved the following interesting results: (3.4) There exists a continuous function f defined on \(<0,1>\), which does not have an approximate derivative at any point of a subset P of \(<0,1>\) of a positive measure and the sets \(\{y\in <0,1>:f(y)>f(x)\}\) and \(\{y\in <0,1>:f(y)<f(x)\}\) are nonporous at any point of P.
(3.6) For any nowhere dense perfect subset P of \(<0,1>\) of a positive measure there exist a continuous function f on \(<0,1>\) and a nonporous path system \(E=\{E_ x:x\in <0,1>\}\) such that \(f_ E'\) exists everywhere in \(<0,1>\) and is in the first class of Baire, \(f'\) exists on \(<0,1>-P\) and it is continuous there, \(f'_{ap}\) exists a.e. in P and the measure of \(\{x\in P:f'_{ap}(x)\neq f_ E'(x)\}\) is positive.
(4.1) For any continuous function f there exists a subset \(A_ f\) of the first category such that at every point \(x\not\in A_ f\) either \((i)\quad S_ x=\{y:y=x,\quad or\quad (f(y)-f(x))/(y-x)\geq 0\}\) is a neighborhood of x, or \((ii)\quad T_ x=\{y:y=x,\quad or\quad 0\geq (f(y)- f(x))/(y-x)\}\) is a neighborhood of x, or (iii) both sets \(S_ x\) and \(T_ x\) have the porosity 1 on both sides at x. Moreover f has a finite derivative at almost every point that is of type (i) or (ii).
(5.4) For every decreasing sequence h of positive numbers converging to zero, the following three assertions are equivalent: (i) For any continuous function f on \(<0,1>\) which has \(f'_{ap}\) everywhere on a measurable set A and \(\bar f{}_ h'\) is on A less than \(\infty\), then \(f'_{ap}=\bar f_ h'\) a.e. on A holds. (ii) For any measurable function f on \(<0,1>\) which has \(f'_{ap}\) everywhere on a measurable set A and \(\bar f{}_ h'\) is less than \(\infty\) on A, then \(f'_{ap}=\bar f_ h'\) a.e. on A holds. (iii) The sequence h has the property (S) of Sindalovskij.
(6.1) For every decreasing sequence of positive numbers \(\{\alpha_ n\}^{\infty}_{n=1}\) converging to zero there exists such a sequence \(h=\{h_ n\}^{\infty}_{n=1}\) of positive numbers converging to zero which does not have the property (S) and for any index k there exists at least one index i for which \(h_ i\in (\alpha_ k,\alpha_{k-1})\).
(7.1) For any sequence of nonzero numbers converging to 0 and for any measurable function f which has an approximate derivative everywhere on a measurable set X, \(f'_{ap}(x)=f_ h'(x)\) at almost every point in X at which \(f_ h'\) exists.