Opuscula Math. 35, no. 6 (2015), 973-978
http://dx.doi.org/10.7494/OpMath.2015.35.6.973
Opuscula Mathematica
Affine extensions of functions with a closed graph
Abstract. Let \(A\) be a closed \(G_{\delta}\)-subset of a normal space \(X\). We prove that every function \(f_0: A\to\mathbb{R}\) with a closed graph can be extended to a function \(f: X\to\mathbb{R}\) with a closed graph, too. This is a consequence of a more general result which gives an affine and constructive method of obtaining such extensions.
Keywords: real-valued functions with a closed graph, points of discontinuity, affine extensions of functions.
Mathematics Subject Classification: 26A15, 54C20, 54D10.
- R.M. Aron, P.D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978) 1, 3-24.
- I. Baggs, Functions with a closed graph, Proc. Amer. Math. Soc. 43 (1974), 439-442.
- J. Borsík, J. Doboš, M. Repický, Sums of quasicontinuous functions with closed graphs, Real Anal. Exch. 25 (1999/2000), 679-690.
- K. Borsuk, Über Isomorphic der Funktionalräume, Bull. Int. Acad. Polon. Sci. (1933), 1-10.
- S. Bromberg, An extension in class \(C^1\), Bol. Soc. Mat. Mex. II, Ser. 27 (1982), 35-44.
- J. Doboš, Sums of closed graph functions, Tatra Mt. Math. Publ. 14 (1998), 9-11.
- J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
- C.L. Fefferman, \(C^m\) extension by linear operators, Annals of Math. 166 (2007) 3, 779-835.
- O.F.K. Kalenda, J. Spurný, Extending Baire-one functions on topological spaces, Topology Appl. 149 (2005) 1-3, 195-216.
- K. Kuratowski, Sur les théorèmes topologiques de la théorie des fonctions de variables réelles, C. R. Acad. Sci. Paris 197 (1933), 19-20.
- J. Merrien, Prolongateurs de fonctions difféntiables d'une variable réelle, J. Math. Pures Appl. 45 (1966) 9, 291-309.
- M. Wójtowicz, W. Sieg, \(P\)-spaces and an unconditional closed graph theorem, RACSAM 104 (2010) 1, 13-18.
- Marek Wójtowicz
- Uniwersytet Kazimierza Wielkiego, Instytut Matematyki, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland
- Waldemar Sieg
- Uniwersytet Kazimierza Wielkiego, Instytut Matematyki, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland
- Communicated by Henryk Hudzik.
- Received: 2014-10-16.
- Accepted: 2014-12-10.
- Published online: 2015-06-06.
