Real Analysis Exchange

On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions

Zbigniew Grande

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In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 933-942.

First available in Project Euclid: 27 June 2008

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

Function with closed graph discrete convergence pointwise convergence transfinite convergence density topology


Grande, Zbigniew. On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions. Real Anal. Exchange 26 (2000), no. 2, 933--942.

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