Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T17:15:19.223Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified approach to continuous and certain non-continuous functions

Part of: Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

J. K. Kohli
J. K. Kohli
Affiliation:
Department of Mathematics Hindu College University of DelhiDelhi-110007, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A unified theory of continuous and certain non-continuous functions is proposed and developed. The proposed theory encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, almost continuous functions, c-continuous functions, c*-continuous functions, s-continuous functions, l-continuous functions, H-continuous functions, and the ε-continuous functions of Klee.

Keywords

P-continuous function almost continuous functionc-continuous functionc*-continuous functionH-continuous functions-continuous function1-continuous functionε-continuous functionsemiconnected functionconnected functionsemilocally Pspacesemilocally connected spacesemiregular spacesaturated spacequasi H-closedH-closed.

MSC classification

Secondary: 54C05: Continuous maps 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 347 - 358
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Bochner, S., ‘Analysis on singularities’, Complex analysis, Vol. II, 1972, (Proc. Conf. Rice Univ., Houston, Texas, 1972, Rice Univ. Studies 59, 1973, No. 2, pp. 21–40).Google Scholar
[2]Bourbaki, N., General topology, (Addison-Wesley, Reading, Massachusetts, 1966).Google Scholar
[3]Gentry, Karl R. and Hoyle, Hughes B. III, ‘c-continuous functions’, Yokohama Math. J. 18 (1970), 71–76.Google Scholar
[4]Hagan, M. R., ‘Conditions for continuity of certain monotone functions’, Proc. Amer. Math. Soc. 30 (1971), 175–178.CrossRefGoogle Scholar
[5]Hwang, Suk Geun, ‘Almost c-continuous functions’, J. Korean Math. Soc. 14 (1978), 229–234.Google Scholar
[6]Jones, John Jr, ‘On semiconnected mappings of topological spaces’, Proc. Amer. Math. Soc. 19 (1968), 174–175.CrossRefGoogle Scholar
[7]Klee, V., ‘Stability of the fixed point property’, Colloq. Math. 8 (1961), 43–46.CrossRefGoogle Scholar
[8]Klee, Victor, and Yandl, Andre, ‘Some proximate concepts in topology’, Symposia Mathematica 16 (1975), 21–39.Google Scholar
[9]Kohli, J. K., ‘A class of mappings containing all continuous and all semiconnected mappings’, Proc. Amer. Math. Soc. 72 (1978), 175–181.CrossRefGoogle Scholar
[10]Kohli, J. K., ‘A unified approach to monotone extensions of mappings’, Acta Math. Acad. Sci. Hungar. 36 (1–2) (1980), 189–194.CrossRefGoogle Scholar
[11]Kohli, J. K., ‘Sufficient conditions for continuity of certain connected functions’, Glasnik Mat. 15 (35) (1980), 377–381.Google Scholar
[12]Kohli, J. K., ‘s-continuous functions and certain weak forms of regularity and complete regularity’, Math. Nachr. 97 (1980), 189–196.CrossRefGoogle Scholar
[13]Kohli, J. K., ‘A class of mappings containing all continuous mappings’, Glasnik Mat. 16 (36) (1981), 361–368.Google Scholar
[14]Lee, Yu-Lee, ‘Some characterizations of semilocally connected spaces’, Proc. Amer. Math. Soc. 16 (1965), 1318–1320.CrossRefGoogle Scholar
[15]Long, Paul E., ‘Concerning semiconnected mappings’, Proc. Amer. Math. Soc. 21 (1969), 117–118.CrossRefGoogle Scholar
[16]Long, Paul E. and Carnahan, Donald A., ‘Comparing almost continuous almost functions’, Proc. Amer. Math. Soc. 38 (1973), 413–418.CrossRefGoogle Scholar
[17]Long, Paul E. and Hamlett, T. R., ‘H-continuous functions’, Boll. Univ. Mat. Ital. 11 (1975), 552–558.Google Scholar
[18]Long, Paul E. and Hendrix, Michael D., ‘Properties of c-continuous functions’, Yokohama Math. J. 22 (1974), 117–123.Google Scholar
[19]Long, Paul E. and Herrington, Larry L., ‘Properties of almost continuous functions’, Boll. Un. Mat. Ital. 10 (1974), 336–342.Google Scholar
[20]Long, Paul E. and Herrington, Larry L., ‘Functions with strongly closed graphs’, Boll. Un. Mat. Ital. 12 (1975), 381–384.Google Scholar
[21]Long, Paul E. and Herrington, Larry L., ‘Properties of c-continuous and c*-continuous functions’, Kyungpook Math. J. 15 (1975), 213–221.Google Scholar
[22]Lorrain, Francois, ‘Notes on topological spaces with minimum neighbourhoods’, Amer. Math. Monthly 76 (1969), 616–627.CrossRefGoogle Scholar
[23]Mägrel, G., ‘A unified approach to measurable and continuous selections’, Trans. Amer. Math. Soc. 248 (1978), 443–452.CrossRefGoogle Scholar
[24]Mathur, Asha, ‘On ∂-continuous mappings’, (preprint).Google Scholar
[25]Mathur, Asha and Deb, Mamta, ‘A note on almost continuous mappings’, Math. Student 40 (1972), 173–184.Google Scholar
[26]Noiri, Takashi, ‘A remark on almost continuous functions’, Proc. Japan Acad. 50 (1974), 205–207.Google Scholar
[27]Noiri, Takashi, ‘Properties of almost c-continuous functions’, J. Korean Math. Soc. 15 (1979), 109–115.Google Scholar
[28]Noiri, Takashi, ‘A generalization of perfect functions’, J. London Math. Soc. 17 (1978), 540–544.CrossRefGoogle Scholar
[29]Noiri, Takashi, ‘Properties of H-continuous functions’, Res. Rep. Yatsushiro Tech. College 1 (1979), 85–90.Google Scholar
[30]Noiri, Takashi, ‘δ-continuous functions’, J. Korean Math. Soc. 16 (1979/1980), 161–166.Google Scholar
[31]Park, Young Soo, ‘c*-continuous functions’, J. Korean Math. Soc. 8 (1971), 69–72.Google Scholar
[32]Porter, J. R. and Thomas, J., ‘On H-closed and minimal Hausdorff spaces’, Trans. Amer. Math. Soc. 138 (1969), 159–170.Google Scholar
[33]Rayburn, Marlon C., ‘Maps and h-normal spaces’, Pacific J. Math. 72 (1979), 549–561.Google Scholar
[34]Sanderson, D. E., ‘Relations among basic properties of non-continuous functions’, Duke Math. J. 38 (1968), 407–414.Google Scholar
[35]Scorborough, C. T. and Stone, A. H., ‘Products of nearly compact spaces’, Trans. Amer. Math. Soc. 124 (1969), 131–147.CrossRefGoogle Scholar
[36]Singal, M. K. and Singal, A. R., ‘Almost continuous mappings’, Yokohama Math. J. 16 (1968), 63–73.Google Scholar
[37]Singal, M. K. and Jam, R. C., ‘Mildly continuous functions’ (Preprint).Google Scholar
[38]Singal, M. K. and Niemse, S. B., ‘z-continuous functions’, Yokohama Math. J., to appear.Google Scholar
[39]Smithson, R. E., ‘A note on δ-continuity and proximate fixed points of multivalued functions’, Proc. Amer. Math. Soc. 23 (1969), 256–260.Google Scholar
[40]Veličko, N. V., ‘H-closed spaces’, Math. Sb. 70 (112) (1966), 98–112 (Russian), translated as Amer. Math. Soc. Transl. (2) 78 (1968), 103–118.Google Scholar