implicit function theorem

implicit function theorem [im′plis·ət ¦fəŋk·shən ‚thir·əm]

(mathematics)

A theorem that gives conditions under which an equation in variablesxandymay be solved so as to expressydirectly as a function ofx; it states that ifF(x,y) and ∂F(x,y)/∂yare continuous in a neighborhood of the point (x₀,y₀) and ifF(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 satisfiesF[x,ƒ(x)] = 0 for |x-x₀| < ε, and satisfies ƒ(x₀) =y₀.