## Taiwanese Journal of Mathematics

### On Totalization of the $H_1$-Integral

Branko Sarić

#### Abstract

Based on the total H$_{1}$-integrability concept, which is established in this paper, we shall try to show that at any point of a compact interval $(a,b]$ in $\mathbb{R}$, at which a point function $F$ has no a discontinuity, $F$ is the total H$_{1}$-indefinite integral of a function $dF_{ex}$ being the limit of $\Delta F_{ex}(I)$, where $I \subseteq [a,b]$, on $[a,b]$, without additional hypotheses on $F$. A residue function of $F$ is introduced. The paper ends with a few of examples that illustrate the theory.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 4 (2011), 1691-1700.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406373

Digital Object Identifier
doi:10.11650/twjm/1500406373

Mathematical Reviews number (MathSciNet)
MR2848983

Zentralblatt MATH identifier
1234.26003

#### Citation

Sarić, Branko. On Totalization of the $H_1$-Integral. Taiwanese J. Math. 15 (2011), no. 4, 1691--1700. doi:10.11650/twjm/1500406373. https://projecteuclid.org/euclid.twjm/1500406373

#### References

• R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Math., Vol. 32, AMS, Providence, 2001.
• I. J. L. Garces\! and P. Y. Lee, Convergence theorems for the H$_{1}$-integral, Taiwanese J. Math., 4(3) (2000), 439-445.
• R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Graduate Studies in Math., Vol. 4, AMS, Providence, 1994.
• D. Hestenes, Multivector Calculus, J. Math. Anal. Appl., 24 (1968), 313-325.
• D. Hestenes, Multivector Functions, J. Math. Anal. Appl., 24 (1968), 467-473.
• A. Macdonald, Stokes' theorem, Real Analysis Exchange, 27 (2002), 739-747.
• J. Mawhin, Generalized Riemann integrals and the divergence theorem for differentiable vector fields, E. B. Christofel, ed., Aachen/Monschau, 1979, Birkhauser, Basel, 1981, pp. 704-714.
• E. J. McShane, Partial Orderings and Moore-Smith Limits, Am. Math. Mon., 59 (1952), 1-11.
• W. Pfeffer, The multidimensional fundamental theorem of calculus, J. Austral. Math. Soc., Ser. A, 43 (1987), 143-170.
• V. Sinha and I. K. Rana, On the continuity of associated interval functions, Real Analysis Exchange 29(2) (2003/2004), 979-981.