Generalizations of the Riemann derivative
J. Marshall Ash · Transactions of the American Mathematical Society · 1967
Introduction.In §1 of this paper a derivative generalizing the Riemann derivative is considered.The existence of this derivative on a set is shown to imply the existence of the Peano derivative almost everywhere on the set.In §2 the W norm (1 ^p 0.We say / is Peano bounded of order k at x, i.e., fe TAx), if there are constants foix),. ..,fk-Ax) such thatLet A={a0, ax,..., ak+l; A0,..., Ak+¡} be a set of real numbers with a^üj if ijtj satisfying 2^ = 0, ; = 0, 1,..., k-1, = k\, j = k.We say that / has a kth generalized derivative with respect to A at the point x, i.e.,fegAx, A), if there is a constant/(W(x)=/(fc)(x, A) such that 2 i = 0 »-ri 2 AJix + att) = fk)ix)tk + oitk) as t -+ 0.