@article{oai:kitami-it.repo.nii.ac.jp:00006402,
author = {ISOBE, Kiro},
issue = {2},
journal = {北見工業大学研究報告},
month = {Jun},
note = {application/pdf, Let f（x） be differentiable, α＜x＜β． We putr(x, y) = { (f(x)?f(y))/(x?y), x≠y, f´(x), ｘ＝ｙ （1）We have the following :(i) r(x, y)＝r(y, x),(ii) r(x, z)(x?z)＝r(x, y)(x?y)＋r(y, z)(y?z),(iii) ___x→y r(x, y)＝r(y, y),(iv) r is continuous at a point (x, y) of x≠y. Newly, having no relation to (1), let r(x, y) be a real function of 2-varia-bles, α＜x, y＜β. In ［1］, if r(x, y) satisfies the previous (i), (ii), (iii), (iv), then we obtain Rolle’s theorem and mean value theorem, namely, (v) if r(a, b)＝0，a＜b, then there exists c(a＜c＜b) such that r(c, c)＝0，(vi) foｒ ａ,ｂ(a＜b), there exists c(a＜c＜b) such that r(a, b)＝r(c, c).The purpose of this note is to point out that if r(x, y) satisfies (i), (ii), (iii), then we have (iv), (v), (vi) and Cauchy's theorem, namely,(vii) if a function s(x, y) satisfies (i), (ii), (iii) and s(x, y)≠0, (a≦x, y≦b) for a＜b, then there exists c(a＜c＜b) such that (r(a, b))/(s(a, b)) = (r(c, c))/(s(c, c)).},
pages = {441--443},
title = {A Certain Consideration on Derivatives and Rolle's Theorem},
volume = {7},
year = {1976}
}