 In general, implicit curves fail the vertical line test (meaning that some values of "x" are associated with more than one value of "y") and so are not necessarily graphs of functions.However, the implicit function theorem gives conditions under which an implicit curve "locally" is given by the graph of a function (so in particular it has no self-intersections).If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.