{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T04:41:38Z","timestamp":1698554498331},"reference-count":15,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2010,11,17]],"date-time":"2010-11-17T00:00:00Z","timestamp":1289952000000},"content-version":"vor","delay-in-days":4338,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1999,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For semi\u2010continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function <jats:italic>f<\/jats:italic> lower semi\u2010computable of type one, if its open hypograph hypo(<jats:italic>f<\/jats:italic>) is recursively enumerably open in dom(<jats:italic>f<\/jats:italic>) \u00d7 \u211d; we call <jats:italic>f<\/jats:italic> lower semi\u2010computable of type two, if its closed epigraph Epi(<jats:italic>f<\/jats:italic>) is recursively enumerably closed in dom(<jats:italic>f<\/jats:italic>) \u00d7 \u211d; we call <jats:italic>f<\/jats:italic> lower semi\u2010computable of type three, if Epi(<jats:italic>f<\/jats:italic>) is recursively closed in dom(<jats:italic>f<\/jats:italic>) \u00d7 \u211d. We show that type one and type two semi\u2010computability are independent and that type three semi\u2010computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi\u2010computable real function on a computable interval is not necessarily computable.<\/jats:p>","DOI":"10.1002\/malq.19990450407","type":"journal-article","created":{"date-parts":[[2010,11,17]],"date-time":"2010-11-17T16:31:05Z","timestamp":1290011465000},"page":"481-496","source":"Crossref","is-referenced-by-count":2,"title":["Approaches to Effective Semi\u2010Continuity of Real Functions"],"prefix":"10.1002","volume":"45","author":[{"given":"Xizhong","family":"Zheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vasco","family":"Brattka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Klaus","family":"Weihrauch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,11,17]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","unstructured":"Brattka V. Computable invariance. In: COCOON '97 Shanghai China August1997 pp.146\u2013155.","DOI":"10.1007\/BFb0045081"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00034-8"},{"key":"e_1_2_1_4_2","unstructured":"Ge X. Private Communication 1997."},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","first-page":"168","DOI":"10.4064\/fm-42-1-168-202","article-title":"Computable functional","volume":"42","author":"Grzegorczyk A.","year":"1955","journal-title":"Fund. Math."},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","first-page":"61","DOI":"10.4064\/fm-44-1-61-71","article-title":"On the definitions of recursive real continuous functions","volume":"44","author":"Grzegorczyk A.","year":"1957","journal-title":"Fund. Math."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21717-7"},{"key":"e_1_2_1_8_2","unstructured":"Richards J. I. andQ.Zhou Computability of closed and open sets in Euclidean space. Preprint1992."},{"key":"e_1_2_1_9_2","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers H.","year":"1967"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69965-8"},{"key":"e_1_2_1_11_2","unstructured":"Weihrauch K. A foundation for computable analysis. In: Combinatorics Complexity Logic (D. S. BRIDGES C. S. CALUDE J. GIBBONS S. REEVES and I. H. WITTEN eds.) Proceedings of DMTCS'96 Springer\u2010Verlag Sigapore1997 pp.66\u201389."},{"key":"e_1_2_1_12_2","unstructured":"Weihrauch K. andV.Brattka Computable Subsets of Euclidean Space. Presented at Workshop of Computa\u2010bility andComplexity in Analysis Dagstuhl Germany April1997. (To appear in Theor. Comput. Sci. 219.)"},{"key":"e_1_2_1_13_2","doi-asserted-by":"crossref","unstructured":"Weihrauch K. andX.Zheng Computability of continuous lower semi\u2010continuous and upper semi\u2010continuous real functions. In: COCOON'97 Shanghai China August1997 pp.166\u2013175. (To appear in Theor. Comput. Sci.211.)","DOI":"10.1007\/BFb0045083"},{"key":"e_1_2_1_14_2","doi-asserted-by":"crossref","unstructured":"Weihrauch K. andX.Zheng Effectivity of the global modulus of continuity on metric spaces. CTCS'97 Santa Margherita Ligure Italy September1997 pp.210\u2013219. (To appear in Theor. Comput. Sci.219.)","DOI":"10.1007\/BFb0026990"},{"key":"e_1_2_1_15_2","doi-asserted-by":"crossref","unstructured":"Weihrauch K. andX.Zheng A finite hierarchy of recursively enumerable real numbers. MFCS'98 Brno Czech Republic August1998 pp.798\u2013806.","DOI":"10.1007\/BFb0055831"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420132"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19990450407","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19990450407","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19990450407","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T21:17:30Z","timestamp":1698527850000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19990450407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,1]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1999,1]]}},"alternative-id":["10.1002\/malq.19990450407"],"URL":"https:\/\/doi.org\/10.1002\/malq.19990450407","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,1]]}}}