Integrability and the Integral of Partial Functions from R into R 1

Noboru Endou; Yasunari Shidama; Masahiko Yamazaki

Formalized Mathematics (2006)

  • Volume: 14, Issue: 4, page 207-212
  • ISSN: 1426-2630

Abstract

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In this paper, we showed the linearity of the indefinite integral [...] the form of which was introduced in [11]. In addition, we proved some theorems about the integral calculus on the subinterval of [a,b]. As a result, we described the fundamental theorem of calculus, that we developed in [11], by a more general expression.

How to cite

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Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. " Integrability and the Integral of Partial Functions from R into R 1 ." Formalized Mathematics 14.4 (2006): 207-212. <http://eudml.org/doc/267262>.

@article{NoboruEndou2006,
abstract = {In this paper, we showed the linearity of the indefinite integral [...] the form of which was introduced in [11]. In addition, we proved some theorems about the integral calculus on the subinterval of [a,b]. As a result, we described the fundamental theorem of calculus, that we developed in [11], by a more general expression.},
author = {Noboru Endou, Yasunari Shidama, Masahiko Yamazaki},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {207-212},
title = { Integrability and the Integral of Partial Functions from R into R 1 },
url = {http://eudml.org/doc/267262},
volume = {14},
year = {2006},
}

TY - JOUR
AU - Noboru Endou
AU - Yasunari Shidama
AU - Masahiko Yamazaki
TI - Integrability and the Integral of Partial Functions from R into R 1
JO - Formalized Mathematics
PY - 2006
VL - 14
IS - 4
SP - 207
EP - 212
AB - In this paper, we showed the linearity of the indefinite integral [...] the form of which was introduced in [11]. In addition, we proved some theorems about the integral calculus on the subinterval of [a,b]. As a result, we described the fundamental theorem of calculus, that we developed in [11], by a more general expression.
LA - eng
UR - http://eudml.org/doc/267262
ER -

References

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  1. [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
  2. [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  3. [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  4. [4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. 
  5. [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
  6. [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  7. [7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  8. [8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990. 
  9. [9] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in ε2. Formalized Mathematics, 6(3):427-440, 1997. 
  10. [10] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999. 
  11. [11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001. 
  12. [12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Integrability of bounded total functions. Formalized Mathematics, 9(2):271-274, 2001. 
  13. [13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001. 
  14. [14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990. 
  15. [15] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990. 
  16. [16] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703-709, 1990. 
  17. [17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990. 
  18. [18] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part I. Formalized Mathematics, 3(1):107-115, 1992. 
  19. [19] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990. 
  20. [20] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990. 
  21. [21] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990. 
  22. [22] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics. 
  23. [23] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990. 
  24. [24] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
  25. [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  26. [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
  27. [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 

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