Real Analysis Exchange

On the Differences of Lower Semicontinuous Functions

Robert Menkyna

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Answering one of the real function problems suggested by A. Maliszewski, the existence of a bounded Darboux function of the Sierpiński first class which cannot be expressed as a difference of two bounded lower semicontinuous functions is proved. As the reply to the other Maliszewski question, we show there exists an almost everywhere continuous Darboux function of the Sierpiński first class which is not a difference of two almost everywhere continuous lower semicontinuous functions.

Article information

Real Anal. Exchange, Volume 41, Number 1 (2016), 123-136.

First available in Project Euclid: 29 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

lower semicontinuity Darboux property Sierpiński function


Menkyna, Robert. On the Differences of Lower Semicontinuous Functions. Real Anal. Exchange 41 (2016), no. 1, 123--136.

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