Unformatted text preview: First we show that Equation ( 3.17 ) determines a well-de±ned function φ ( x ): For notational convenience, we assume without loss of generality that f ( x ,y ) = 0 (that is, c = 0), and ∂f ∂y ( x ,y ) > . Since f ( x,y ) is continuous, we know that ∂f ∂y ( −→ x ) > 0 at all points −→ x = ( x,y ) su²ciently near −→ x , say for | x − x | ≤ δ and | y − y | ≤ δ 2 . For any a ∈ [ x − δ,x + δ ], consider the function of y obtained by ±xing the value of x at x = a : g a ( y ) = f ( a,y ) ;...
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- Fall '08
- Calculus, Continuous function, implicit function theorem, −, continuously differentiable function