implicit function theorem

implicit function theorem

[im′plis·ət ¦fəŋk·shən ‚thir·əm]

(mathematics)

A theorem that gives conditions under which an equation in variables x and y may be solved so as to express y directly as a function of x ; it states that if F (x,y) and ∂ F (x,y)/∂ y are continuous in a neighborhood of the point (x₀, y₀) and if F (x,y) = 0 and ∂ F (x,y)/∂ y ≠ 0, then there is a number ε > 0 such that there is one and only one function ƒf(x) that is continuous and satisfies F [x,ƒ(x)] = 0 for | x-x₀| < ε,="" and="" satisfies="">x₀) = y₀.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.