Isolated Point Singularities on Curves

Question

I know that a curve $C: f(x, y) = 0$ is singular provided there is a point $(x_0, y_0) \in C$ for which $$\dfrac{\partial f}{\partial x}(x_0, y_0)=\dfrac{\partial f}{\partial y}(x_0, y_0)=0.$$ However, suppose the singular point is an isolated point. For example, the point $(-1,0)$ on the curve $$C:y^2=x^3-3x-2=(x-2)(x+1)^2.$$ Both partials are $0$ when evaluated at $(-1, 0)$, but if the definition of a partial derivative is defined using a limit and the curve $C$ is not defined in any neighborhood around the point $(-1, 0)$, don't the partials fail to exist? Any insight would be appreciated.