The Least Root of a Continuous Function

Abstract

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