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                      

#### 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

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