Journal of Formalized Mathematics
Volume 12, 2000
University of Bialystok
Copyright (c) 2000 Association of Mizar Users

## Definition of Integrability for Partial Functions from \$\Bbb R\$ to \$\Bbb R\$ and Integrability for Continuous Functions

Noboru Endou
Shinshu University, Nagano
Katsumi Wasaki
Shinshu University, Nagano
Yasunari Shidama
Shinshu University, Nagano

### Summary.

In this article, we defined the Riemann definite integral of partial function from \${\Bbb R}\$ to \${\Bbb R}\$. Then we have proved the integrability for the continuous function and differentiable function. Moreover, we have proved an elementary theorem of calculus.

#### MML Identifier: INTEGRA5

The terminology and notation used in this paper have been introduced in the following articles [19] [21] [1] [20] [9] [3] [22] [4] [18] [7] [2] [12] [13] [5] [11] [10] [17] [15] [6] [8] [16] [14]

#### Contents (PDF format)

1. Some Useful Properties of Finite Sequence
2. Integrability for Partial Function of \${\Bbb R}\$, \${\Bbb R}\$
3. Integrability for Continuous Function

#### Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[6] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[7] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in \$\calE^2\$. Journal of Formalized Mathematics, 9, 1997.
[8] Noboru Endou and Artur Kornilowicz. The definition of the Riemann definite integral and some related lemmas. Journal of Formalized Mathematics, 11, 1999.
[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[10] Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.
[11] Jaroslaw Kotowicz. Convergent sequences and the limit of sequences. Journal of Formalized Mathematics, 1, 1989.
[12] Jaroslaw Kotowicz. Real sequences and basic operations on them. Journal of Formalized Mathematics, 1, 1989.
[13] Jaroslaw Kotowicz. Partial functions from a domain to the set of real numbers. Journal of Formalized Mathematics, 2, 1990.
[14] Jaroslaw Kotowicz. Properties of real functions. Journal of Formalized Mathematics, 2, 1990.
[15] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[16] Konrad Raczkowski and Pawel Sadowski. Real function continuity. Journal of Formalized Mathematics, 2, 1990.
[17] Konrad Raczkowski and Pawel Sadowski. Real function differentiability. Journal of Formalized Mathematics, 2, 1990.
[18] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[19] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[20] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[21] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.