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

Jaroslaw Kotowicz

Warsaw University, Bialystok

Supported by RPBP.III24.C8.
Summary.

We introduce the leftside and the rightside
limit of a real function at a point. We prove a few properties of the operations
on the proper and improper oneside limits and show that Cauchy and Heine
characterizations of the oneside limit are equivalent.
The terminology and notation used in this paper have been
introduced in the following articles
[11]
[13]
[2]
[12]
[3]
[1]
[9]
[5]
[4]
[14]
[10]
[7]
[8]
[6]
Contents (PDF format)
Bibliography
 [1]
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]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Jaroslaw Kotowicz.
The limit of a real function at infinity.
Journal of Formalized Mathematics,
2, 1990.
 [7]
Jaroslaw Kotowicz.
Partial functions from a domain to the set of real numbers.
Journal of Formalized Mathematics,
2, 1990.
 [8]
Jaroslaw Kotowicz.
Properties of real functions.
Journal of Formalized Mathematics,
2, 1990.
 [9]
Jan Popiolek.
Some properties of functions modul and signum.
Journal of Formalized Mathematics,
1, 1989.
 [10]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
Journal of Formalized Mathematics,
2, 1990.
 [11]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [12]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
 [13]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [14]
Edmund Woronowicz.
Relations defined on sets.
Journal of Formalized Mathematics,
1, 1989.
Received August 20, 1990
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