Real Analysis Exchange

Monotone and Discrete Limits of Continuous Functions

Vilmos Prokaj

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In this note we prove that for a quite large class of topological spaces every upper semi-continuous function, which is a discrete limit of continuous functions, it is also a pointwise decreasing discrete limit of continuous functions. This question was motivated by a paper of Zbigniew Grande. He asked that whether it be true for the topology of right hand continuity on the real line. He gave a partial answer showing that under some additional condition imposed on the function the answer is affirmative.

Article information

Real Anal. Exchange, Volume 25, Number 2 (1999), 879-886.

First available in Project Euclid: 3 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A03: Foundations: limits and generalizations, elementary topology of the line 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] 26A99: None of the above, but in this section

upper-semi continuous function discrete limit


Prokaj, Vilmos. Monotone and Discrete Limits of Continuous Functions. Real Anal. Exchange 25 (1999), no. 2, 879--886.

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  • Zbigniew Grande, On A.C. Limits and Monotone Limits of Sequences of Jump Functions, Real Analysis Exchange, 25.2(2000), –. appear).
  • Ákos Császár and Miklós Laczkovich, Discrete and Equal Baire Classes, Acta Math. Hung., 55(1990), 165–178.