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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
https://kitami-it.repo.nii.ac.jp/records/6417
https://kitami-it.repo.nii.ac.jp/records/6417fe049e62-b475-4ddb-9a26-ebc3a1de3335
名前 / ファイル | ライセンス | アクション |
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8-1-13.pdf (2.3 MB)
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Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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公開日 | 2007-04-09 | |||||
タイトル | ||||||
タイトル | 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 | |||||
言語 | ||||||
言語 | jpn | |||||
資源タイプ | ||||||
資源 | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | departmental bulletin paper | |||||
その他のタイトル | ||||||
その他のタイトル | On Differential Coefficients and Mean values : Some Notes on Differentiability of Real Functions | |||||
著者 |
磯部, 煕郎
× 磯部, 煕郎 |
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著者別名 | ||||||
識別子 | 32512 | |||||
識別子Scheme | WEKO | |||||
姓名 | ISOBE, Kiro | |||||
抄録 | ||||||
内容記述タイプ | Abstract | |||||
内容記述 | 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. | |||||
書誌情報 |
北見工業大学研究報告 巻 8, 号 1, p. 133-139, 発行日 1976-11 |
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フォーマット | ||||||
内容記述タイプ | Other | |||||
内容記述 | application/pdf | |||||
著者版フラグ | ||||||
値 | publisher | |||||
出版者 | ||||||
出版者 | 北見工業大学 |