The Least Root of a Continuous Function
- 10 Downloads
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 phrasesImplicit functions continuousness zeros of functions
Unable to display preview. Download preview PDF.
- 3.V. A. Il’in and E. G. Poznyak, Fundamentals ofMathematical Analysis, Part 1, 7th ed. (Moscow, Fizmatlit, 2005) [in Russian].Google Scholar
- 4.V. D. Matrosov, R. M. Aslanov, and M. V. Topunov, Differential Equations and Partial Differential Equations (Vlados, Moscow, 2011) [in Russian].Google Scholar
- 7.B. P. Demidovich, Approximation Methods (Nauka, Moscow, 2015) [in Russian].Google Scholar
- 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.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