## Real Analysis Exchange

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

Zbigniew Grande

#### Abstract

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

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

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571383

Mathematical Reviews number (MathSciNet)
MR1844409

Zentralblatt MATH identifier
1024.26003

#### Citation

Grande, Zbigniew. On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions. Real Anal. Exchange 26 (2000), no. 2, 933--942. https://projecteuclid.org/euclid.rae/1214571383

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