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2009/2010 On Approximations of Semicontinuous Functions by Darboux Semicontinuous Functions
Robert Menkyna
Real Anal. Exchange 35(2): 423-430 (2009/2010).


In the whole article, the function is understood as the real valued function of real variables. It is well known that the construction of semicontinuous functions is possible either as a limit (in the sense of pointwise convergence) of the monotonous sequence of continuous functions or with assistance of the certain system of associated sets. In the first case a good understanding of the constructed function is obtained. However in the second case the Darboux property of semicontinuous function is assured when certain requirements on the system of associated sets are satisfied. Therefore the combination of both approaches seems to be the optimal method when a semicontinuous function with Darboux property is constructed. Such a combined method is used when the main theorem of this paper is proven. Later, the validity of the main theorem helps us to address the problem published by J. G. Ceder and T. L. Pearson in [2].


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Robert Menkyna. "On Approximations of Semicontinuous Functions by Darboux Semicontinuous Functions." Real Anal. Exchange 35 (2) 423 - 430, 2009/2010.


Published: 2009/2010
First available in Project Euclid: 22 September 2010

zbMATH: 1222.26006
MathSciNet: MR2683608

Primary: 26A15 , 26A21

Keywords: Baire 1 functions , lower resp. upper semicontinuous functions , the Darboux property

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 2 • 2009/2010
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