In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be wellbehaved in some particular way, such as differentiability. See Singularity theory for general discussion of the geometric theory, which only covers some aspects.
For example, the function
on the real line has a singularity at x = 0, where it seems to "explode" to ±∞ and is not defined. The function g(x) = x (see absolute value) also has a singularity at x = 0, since it is not differentiable there. Similarly, the graph defined by y^{2} = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.
The algebraic set defined by y^{2} = x^{2} in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.
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In real analysis singularities are also called discontinuities. There are three kinds: type I, which has two subtypes, and type II, which also can be divided into two subtypes, but normally is not.
To describe these types, suppose that f(x) is a function of a real argument x, and for any value of its argument, say c, the symbols f(c ^{+} ) and f(c ^{−} ) are defined by:
The limit f(c ^{−} ) is called the lefthanded limit, and f(c ^{+} ) is called the righthanded limit. The value f(c ^{−} ) is the value that the function f(x) tends towards as the value x approaches c from below, and the value f(c ^{+} ) is the value that the function f(x) tends towards as the value x approaches c from above, regardless of the actual value the function has at the point where x = c .
There are some functions for which these limits do not exist at all. For example the function
does not tend towards anything as x approaches c = 0. The limits in this case are not infinite, but rather undefined: there is no value that g(x) settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.
In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.
A coordinate singularity (or coördinate singularity) occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity.
In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, and the point a is an element of U, and f is a complex differentiable function defined on some neighborhood around a, excluding a: U \ {a}.
A finitetime singularity occurs when a kinematic variable increases towards infinity at a finite time. An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finitetime singularities include Euler's disk, the Painlevé paradox, and Heinz von Foerster's Doomsday's Equation.
In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a point of a scheme whose stalk is not a regular local ring). For example, y^{2} − x^{3} = 0 defines an isolated singular point (at the cusp) x = y = 0. The ring in question is given by
The maximal ideal of the localization at (t^{2},t^{3}) is a height one local ring generated by two elements and thus not regular.
