## Real Analysis Exchange

### Generalized continuity and uniform approximation by step functions.

Christian Richter

#### Abstract

Given two topological spaces $X$ and $Y$ and a family ${\mathcal O}_\ast$ of subsets of $X$, a function $f: X \rightarrow Y$ is called ${\mathcal O}_\ast$-continuous if $f^{-1}(V) \in {\mathcal O}_\ast$ for every open set $V \subseteq Y$. An ${\mathcal O}_\ast$-step function is meant to be a function $\varphi: X \rightarrow Y$ that is piecewise constant on a partition of $X$ into sets from ${\mathcal O}_\ast$. Using some technical assumptions on $X$, $Y$, and ${\mathcal O}_\ast$ we give representations of ${\mathcal O}_\ast$-continuous functions as uniform limits of ${\mathcal O}_\ast$-step functions. We deal in particular with $\alpha$-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions.

#### Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 215-238.

Dates
First available in Project Euclid: 5 June 2006

https://projecteuclid.org/euclid.rae/1149516808

Mathematical Reviews number (MathSciNet)
MR2218199

Zentralblatt MATH identifier
1101.54017

Subjects
Primary: 54C08: Weak and generalized continuity
Secondary: 41A30: Approximation by other special function classes

#### Citation

Richter, Christian. Generalized continuity and uniform approximation by step functions. Real Anal. Exchange 31 (2005), no. 1, 215--238. https://projecteuclid.org/euclid.rae/1149516808

#### References

• M. E. Abd El-Monsef, S. N. El-Deeb, R. A. Mahmoud, $\beta$-open sets and $\beta$-continuous mapping, Bull. Fac. Sci. Assiut Univ. A, 12 (1983), 77–90.
• D. Andrijević, Semi-preopen sets, Mat. Vesnik, 38 (1986), 24–32.
• W. W. Bledsoe, Neighborly functions, Proc. Amer. Math. Soc., 3 (1952), 114–115.
• H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc., 24 (1922), 113–128.
• J. Borsík, J. Doboš, On decompositions of quasicontinuity, Real Anal. Exchange, 16 (1990/91), 292–305.
• H. H. Corson, E. Michael, Metrizability of certain countable unions, Illinois J. Math., 8 (1964), 351–360.
• R. Engelking, General topology, Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, 1989.
• Z. Frol\' \ik, Remarks concerning the invariance of Baire spaces under mappings, Czechoslovak Math. J., 11 (86) (1961), 381–385.
• K. R. Gentry, H. B. Hoyle, III, Somewhat continuous functions, Czechoslovak Math. J., 21 (96) (1971), 5–12.
• T., Husain, Almost continuous mappings, Prace Mat., 10 (1966), 1–7.
• S. Kempisty, Sur les fonctions quasicontinues, Fund. Math., 19 (1932), 184–197.
• N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41.
• A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deep, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53 (1983).
• A. S. Mashhour, I. A. Hasanein, S. N. El-Deeb, $\alpha$-continuous and $\alpha$-open mappings, Acta Math. Hungar., 41 (1983), 213–218.
• S. A. Naimpally, Review on [richter/stephani2003?] in Mathematical Reviews (MR2061313).
• O. Njåstad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961–970.
• Z. Piotrowski, Some remarks on almost continuous functions, Math. Slovaca, 39 (1989), 75–80.
• V. Popa, Asupra unor forme slăbite de continuitate (On certain weakened forms of continuity), Stud. Cerc. Mat., 33 (1981), 543–546.
• V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France, 86 (1958), 41–74.
• C. Richter, I. Stephani, Cluster sets and approximation properties of quasi-continuous and cliquish functions, Real Anal. Exchange, 29 (2003/04), 299–321.
• S. Z. Shi, Q. Zheng, D. Zhuang, Discontinuous robust mappings are approximatable, Trans. Amer. Math. Soc., 347 (1995), 4943–4957.
• J. Smítal, E., Stanová, On almost continuous functions, Acta Math. Univ. Comenian., 37 (1980), 147–155.
• M. Wilhelm, On a question of B.J. Pettis, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 591–592 (1980).
• Q. Zheng, Robust analysis and global minimization of a class of discontinuous functions (I), Acta Math. Appl. Sinica (English Ser.), 6 (1990), 205–223.