@article{oai:kitami-it.repo.nii.ac.jp:00006417,
 author = {磯部, 煕郎},
 issue = {1},
 journal = {北見工業大学研究報告},
 month = {Nov},
 note = {application/pdf, Let f (x) be a Darboux function on [a, b] and f (a)=f(b). In this paper first we show that for any ε＞O there exist c and d such as a＜c＜d＜b, d−c＜ε and f(c)=f(d). Some methods for proof of this proposition which we use essentially are due to [1].　　In the next place, let f(x) be continuous on [a, b].　Adapting the previous proposition to f(x), we have the following proposition that for any ε＞O there exist c and d such as a＜c＜d＜b, d−ｃ＜ε and ( f(d)−f(c))/(d−ｃ) ＝ (f(b)−f(a))/(b−c).　Consequently we can show that there exist two sequences ｛x_n｝ and {y_n} such as a＜c_1＜c_2＜…＜d_2＜d_1＜b, d_n−c_n→0 and　　　　　　(f(d_n)一f(c_n))/(d_n−c_n) = (f(b)−f(a))/(b−a) Thus we obtain x_0 in(ａ, b) as <lim>___<n→＋∞> c_n ＝ <lim>___<n→＋∞> d_n＝x_0. Moreover, let f(x) be continuous on (α，β) and we put Γ＝{γ：γ = (f(b)−f(a))/(b−a) , a, b∈(α, β) and a≠b }. To any ε＞O there exist two sequences {x_n} and {y_n} such that α＜x_1＜x_2＜…＜y_2＜y_1＜β， y_n−x_n →0 and　　(f(y_n)−ｆ（x_n）)/(y_n−x_n) = γ and we obtain ｘ in (α,β)as <lim>___<n→＋∞> x_n＝<lim>___<n→＋∞> y_n=x.　　　Totality of all such x is denoted X_γ（γ∈Γ）　and to any ｘ(αくxくβ) we put Γ＝{γ：x∈X_γ}.　The following theorems are established: Theorem 1. If f(x) is differentiable at x_0(αくx_0くβ) and Γ_(x_0)≠Φ, thenΓ_(x_0) = ｛ｆ´(x)｝. Theorem 2. IfΓ_(x_0) = ｛γ｝, then f(x) is differentiable at x_0 and f´(x_0) = γ.},
 pages = {133--139},
 title = {微分係数と平均値について　：　実関数の微分可能性についての一考察},
 volume = {8},
 year = {1976}
}