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A Certain Consideration on Derivatives and Rolle's TheoremISOBE, Kiroapplication/pdfLet 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)　<lim>___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)).北見工業大学1976-06engdepartmental bulletin paperhttps://kitami-it.repo.nii.ac.jp/records/6402北見工業大学研究報告72441443https://kitami-it.repo.nii.ac.jp/record/6402/files/7-2-13.pdfapplication/pdf1.1 MB2016-11-22