Given $f(x,y)$ continuously differentiable in a domain of the plane, $f(a,b) = 0$ and $f_x(a,b)\ne 0$. Can you show $f(x,y) = 0$ in a finite neighborhood of $(a,b)$?
Examples
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Sphere tangent to the plane.
$f(a,b )= 0$ at one point but $f_x(a,b) = 0$ and $f_y(a,b) = 0$ and there is no neighborhood of $(a,b)$ in which $f(x,y)=0$.
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Sphere intersecting the plane.
There are points $(a,b)$ (on the circle of intersection) where $f(a,b) =0 $ , $f_x(a,b)\ne 0$ , and $f(x,y)=0$ in a neighborhood of $(a,b)$, (circular arc through $(a,b)$ )
But how do you show this in general without geometry?
Application: Implicit Function Theorem.
Best Answer
Utterly false: $$f(x,y) = x,\qquad f(0,0) = 0,\qquad f_x(0,0) = 1\ne 0.$$