Real Analysis Exchange

Sums of Quasicontinuous Functions with Closed Graphs

Ján Borsík, Jozef Doboš, and Miroslav Repický

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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

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

First available in Project Euclid: 3 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C08: Weak and generalized continuity
Secondary: 54C30: Real-valued functions [See also 26-XX]

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


Borsík, Ján; Doboš, Jozef; Repický, Miroslav. Sums of Quasicontinuous Functions with Closed Graphs. Real Anal. Exchange 25 (1999), no. 2, 679--690.

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