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.

单词	implicit function theorem
释义	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.
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