## Real Analysis Exchange

- Real Anal. Exchange
- Volume 25, Number 2 (1999), 679-690.

### Sums of Quasicontinuous Functions with Closed Graphs

Ján Borsík, Jozef Doboš, and Miroslav Repický

#### Abstract

We prove that every real-valued $\mathcal{B}^*_1$ function $f$ defined on a~separable metric space $X$ is the sum of three quasicontinuous functions with closed graphs, and there is a $\mathcal{B}^*_1$ function which is not the sum of two quasicontinuous functions with closed graphs. Consequently, if $X$ is a separable metric space which is a Baire space in the strong sense, then the next three properties are equivalent: (1)$f$ is a $\mathcal{B}^*_1$ function, (2) $f$ is the sum of (at least) three quasicontinuous functions with closed graphs, and (3) $f$ is a piecewise continuous function.

#### Article information

**Source**

Real Anal. Exchange Volume 25, Number 2 (1999), 679-690.

**Dates**

First available in Project Euclid: 3 January 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1230995403

**Mathematical Reviews number (MathSciNet)**

MR1778521

**Zentralblatt MATH identifier**

1021.54015

**Subjects**

Primary: 54C08: Weak and generalized continuity

Secondary: 54C30: Real-valued functions [See also 26-XX]

**Keywords**

Quasicontinuous function piecewise continuous function function of the class~$\baire$ function with closed graph

#### Citation

Borsík, Ján; Doboš, Jozef; Repický, Miroslav. Sums of Quasicontinuous Functions with Closed Graphs. Real Anal. Exchange 25 (1999), no. 2, 679--690. https://projecteuclid.org/euclid.rae/1230995403.