Web Reference: The purpose of the implicit function theorem is to tell us that functions like g1(x) and g2(x) almost always exist, even in situations where we cannot write down explicit formulas. 1 The Implicit Function Theorem Suppose that (a; b) is a point on the curve F(x; y) = 0 where and suppose that this equation can be solved for y as a function of x for all (x; y) sufficiently near (a; b). The implicit function theorem gives conditions under which it is possible to solve for x as a function of p in the neighborhood of a known solution ( ̄x, ̄p). There are actually many implicit function theorems. If you make stronger assumptions, you can derive stronger conclusions.
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