書誌事項
- タイトル別名
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- ビブン ケイスウ ト ヘイキンチ ニ ツイテ ジツカンスウ ノ ビブン カノウ
- On Differential Coefficients and Mean values : Some Notes on Differentiability of Real Functions
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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.
収録刊行物
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- 北見工業大学研究報告
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北見工業大学研究報告 8 (1), 133-139, 1976-11
北見工業大学
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詳細情報 詳細情報について
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- CRID
- 1050564288760416256
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- NII論文ID
- 120000786489
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- NII書誌ID
- AN00052475
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- ISSN
- 03877035
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- NDL書誌ID
- 1798239
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- 本文言語コード
- ja
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- 資料種別
- departmental bulletin paper
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- データソース種別
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- IRDB
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