A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field K is taken as the reals). More formally, a point (a, b) on a curve is singular if the x and y partial derivatives of f are both zero at the point (a, b). (If the field K is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)
Consider the following two examples. For the curve

the cusp at (0, 0) is a singular point. For the curve

is a nonsingular point and this curve is nonsingular.
For a second-order ordinary differential equation, consider

If P(x) and Q(x) remain finite at , then is called an ordinary point . If either P(x) or Q(x) diverges as http://mathworld.wolfram.com/s1img1155.gif , then is called a singular point. Singular points are further classified as follows:
1. If either P(x) or Q(x) diverges as but and remain finite as , then is called a regular singular point (or nonessential singularity).
2. If P(x) diverges more quickly than , so approaches infinity as , or Q(x) diverges more quickly than so that goes to infinity as , then is called an irregular singularity (or essential singularity).
Singular points are sometimes known as singularities, and vice versa.