{"created":"2021-03-01T05:58:51.084989+00:00","id":6417,"links":{},"metadata":{"_buckets":{"deposit":"f564af26-cbd5-4121-962b-81d42d8dcb5c"},"_deposit":{"id":"6417","owners":[],"pid":{"revision_id":0,"type":"depid","value":"6417"},"status":"published"},"_oai":{"id":"oai:kitami-it.repo.nii.ac.jp:00006417","sets":["7:15:29"]},"item_2_alternative_title_18":{"attribute_name":"その他のタイトル","attribute_value_mlt":[{"subitem_alternative_title":"On Differential Coefficients and Mean values : Some Notes on Differentiability of Real Functions"}]},"item_2_biblio_info_6":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicIssueDates":{"bibliographicIssueDate":"1976-11","bibliographicIssueDateType":"Issued"},"bibliographicIssueNumber":"1","bibliographicPageEnd":"139","bibliographicPageStart":"133","bibliographicVolumeNumber":"8","bibliographic_titles":[{"bibliographic_title":"北見工業大学研究報告"}]}]},"item_2_description_14":{"attribute_name":"フォーマット","attribute_value_mlt":[{"subitem_description":"application/pdf","subitem_description_type":"Other"}]},"item_2_description_4":{"attribute_name":"抄録","attribute_value_mlt":[{"subitem_description":"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.","subitem_description_type":"Abstract"}]},"item_2_full_name_3":{"attribute_name":"著者別名","attribute_value_mlt":[{"nameIdentifiers":[{"nameIdentifier":"32512","nameIdentifierScheme":"WEKO"}],"names":[{"name":"ISOBE, Kiro"}]}]},"item_2_publisher_32":{"attribute_name":"出版者","attribute_value_mlt":[{"subitem_publisher":"北見工業大学"}]},"item_2_select_15":{"attribute_name":"著者版フラグ","attribute_value_mlt":[{"subitem_select_item":"publisher"}]},"item_creator":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"磯部, 煕郎"}],"nameIdentifiers":[{"nameIdentifier":"32511","nameIdentifierScheme":"WEKO"}]}]},"item_files":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2016-11-22"}],"displaytype":"detail","filename":"8-1-13.pdf","filesize":[{"value":"2.3 MB"}],"format":"application/pdf","licensetype":"license_note","mimetype":"application/pdf","url":{"label":"8-1-13.pdf","url":"https://kitami-it.repo.nii.ac.jp/record/6417/files/8-1-13.pdf"},"version_id":"59c53d39-f450-474f-ae2a-b9ed03379a85"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"jpn"}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourcetype":"departmental bulletin paper","resourceuri":"http://purl.org/coar/resource_type/c_6501"}]},"item_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","item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_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"}]},"item_type_id":"2","owner":"1","path":["29"],"pubdate":{"attribute_name":"公開日","attribute_value":"2007-04-09"},"publish_date":"2007-04-09","publish_status":"0","recid":"6417","relation_version_is_last":true,"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"],"weko_creator_id":"1","weko_shared_id":3},"updated":"2022-12-13T02:23:38.608479+00:00"}