the definition of a singular point is local, the definition given for an affine analytic/algebraic variety applies to the points of any complex analytic or algebraic variety. The set of points of a variety which are singular is called the singular locus, denoted , and the complement is the non-singular or smooth locus, denoted . We say a complex variety is smooth or non-singular if it's singular locus is empty. That is, if it is equal to its non-singular locus.

By the implicit function theorem