Lobachevskii Journal of Mathematics

, Volume 39, Issue 2, pp 200–203 | Cite as

The Least Root of a Continuous Function

Article
  • 10 Downloads

Abstract

For each ε > 0 and each scalar real valued and continuous on a compact set Ω ⊂ R n , ξ ∈ [a, b] function g(τ, ξ) such that g(τ, a) · g(τ, b) < 0 we construct a function gε(τ, ξ), for which the least root ξ(τ) of the equation gε(τ, ξ) = 0 continuously depends on τ, while |g(τ, ξ) − gε(τ, ξ)| < ε. We give examples illustrating the fact that in a general case assumptions are unimprovable.

Keywords and phrases

Implicit functions continuousness zeros of functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Bolzano, Rein analytischer Beweis des Lehrsatzes, daßzwischen je zweiWerthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Haase, Prag, 1817) [in German].MATHGoogle Scholar
  2. 2.
    E. Kol’man, Bernard Bolzano (Akad. Nauk SSSR, Moscow, 1955) [in Russian].MATHGoogle Scholar
  3. 3.
    V. A. Il’in and E. G. Poznyak, Fundamentals ofMathematical Analysis, Part 1, 7th ed. (Moscow, Fizmatlit, 2005) [in Russian].Google Scholar
  4. 4.
    V. D. Matrosov, R. M. Aslanov, and M. V. Topunov, Differential Equations and Partial Differential Equations (Vlados, Moscow, 2011) [in Russian].Google Scholar
  5. 5.
    Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, Optimal Control. Linear Theory and Applications (Mosk. Gos. Univ., Moscow, 2007) [in Russian].MATHGoogle Scholar
  6. 6.
    E. M. Karchevskii and M. M. Karchevskii, Lectures in Linear Algebra and Analytical Geometry (Kazan Gos. Univ., Kazan, 2014) [in Russian].MATHGoogle Scholar
  7. 7.
    B. P. Demidovich, Approximation Methods (Nauka, Moscow, 2015) [in Russian].Google Scholar
  8. 8.
    V. S. Mokeichev, in Contemporary Problems of the Theory of Functions and Their Applications, Proceedings of the 16th Saratov Winter School (Saratov Gos. Univ., Saratov, 2012), pp. 122–123.Google Scholar
  9. 9.
    V. S. Mokeichev and I. E. Filippov, “Dependence of the least root of a continuous function on a parameter,” in Proceedings of the International Conference on Algebra, Analysis, and Geometry (Kazan Gos. Univ., Kazan, 2016), pp. 248–249.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazan, TatarstanRussia

Personalised recommendations