Singularity (mathematics)

In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the function has a singularity at , where the value of the function is not defined, as involving a division by zero. The absolute value function also has a singularity at , since it is not differentiable there. The algebraic curve defined by in the coordinate system has a singularity (called a cusp) at . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not). To describe the way these two types of limits are being used, suppose that is a function of a real argument , and for any value of its argument, say , then the left-handed limit, , and the right-handed limit, , are defined by: constrained by and constrained by . The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where . There are some functions for which these limits do not exist at all. For example, the function does not tend towards anything as approaches . The limits in this case are not infinite, but rather undefined: there is no value that settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity. The possible cases at a given value for the argument are as follows. A point of continuity is a value of for which , as one expects for a smooth function. All the values must be finite.