The equation f(x,y) = x² + y² – 1 = 0 defines y implicitly in terms of x. At any point of the circle, except (±1,0), the curve can be locally defined as a graph, either $y=g(x)=\sqrt{1-x^2}$ or $y=g(x)=-\sqrt{1-x^2.}$ But this is not possible when x = ±1, as the derivative dx/dy is zero. More generally, when a dependent variable y is implicitly related to independent variables x₁,x₂,…,x_k by f(x₁,x₂,…,x_k,y) = 0, then y can be written as a graph y = g(x₁,x₂,…,x_k), local to a point p, if ∂f/∂y(p) ≠ 0. The theorem more generally applies when m dependent variables y_i are implicitly defined in terms of n independent variables x_i by m equations f_i, in which case the Jacobian (∂f_i/∂y_j) must be invertible.