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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 |
|---|---|---|
8-1-13.pdf (2.3 MB)
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| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
|---|---|---|---|---|---|---|
| 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 | ja | |||||
| 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 | |||||
| 言語 | en | |||||
| 著者 |
磯部, 煕郎
× 磯部, 煕郎 |
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| 著者別名 | ||||||
| 識別子Scheme | WEKO | |||||
| 識別子 | 32512 | |||||
| 姓名 | ISOBE, Kiro | |||||
| 言語 | en | |||||
| 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. | |||||
| 言語 | en | |||||
| 書誌情報 |
ja : 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 | ||||||
| 言語 | en | |||||
| 値 | publisher | |||||
| 出版者 | ||||||
| 出版者 | 北見工業大学 | |||||
| 言語 | ja | |||||
