Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990
Association of Mizar Users
The Limit of a Real Function at a Point

Jaroslaw Kotowicz

Warsaw University, Bialystok

Supported by RPBP.III24.C8.
Summary.

We define the proper and the improper limit
of a real function at a point.
The main properties of the operations on the limit
of function are proved. The connection between the oneside
limits and the limit of function at a point are exposed.
Equivalent Cauchy and Heine characterizations of the limit
of real function at a point are proved.
The terminology and notation used in this paper have been
introduced in the following articles
[13]
[15]
[2]
[14]
[4]
[1]
[16]
[3]
[11]
[6]
[5]
[12]
[9]
[10]
[7]
[8]
Contents (PDF format)
Bibliography
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Grzegorz Bancerek.
The fundamental properties of natural numbers.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Grzegorz Bancerek.
The ordinal numbers.
Journal of Formalized Mathematics,
1, 1989.
 [3]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
Journal of Formalized Mathematics,
1, 1989.
 [7]
Jaroslaw Kotowicz.
The limit of a real function at infinity.
Journal of Formalized Mathematics,
2, 1990.
 [8]
Jaroslaw Kotowicz.
The oneside limits of a real function at a point.
Journal of Formalized Mathematics,
2, 1990.
 [9]
Jaroslaw Kotowicz.
Partial functions from a domain to the set of real numbers.
Journal of Formalized Mathematics,
2, 1990.
 [10]
Jaroslaw Kotowicz.
Properties of real functions.
Journal of Formalized Mathematics,
2, 1990.
 [11]
Jan Popiolek.
Some properties of functions modul and signum.
Journal of Formalized Mathematics,
1, 1989.
 [12]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
Journal of Formalized Mathematics,
2, 1990.
 [13]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
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 [14]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
 [15]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [16]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received September 5, 1990
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