Thus, here, the "Y" in the statement of the theorem is just the number 2"b"; the linear map defined by it is invertible iff "b" ≠ 0. By the implicit function theorem we see that we can locally write the circle in the form "y" = "g"("x") for all points where "y" ≠ 0. For (±1, 0) we run into trouble, as noted before. The implicit function theorem may still be applied to these two points, by writing "x" as a function of "y", that is, formula_41; now the graph of the function will be formula_42, since where "b = 0" we have "a = 1", and the conditions to locally express the function in this form are satisfied.