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 well-behaved 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

f(x) = 1/x

on the real line has a singularity at x = 0, where it explodes to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. Similarly, the graph defined by y² = 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 y² = x² in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

Complex analysis

In complex analysis, there are four kinds of singularity. Suppose U is an open subset of C, a is an element of U and f is a holomorphic function defined on U \ {a}.

• The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}.

• The point a is a pole of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z − a)ⁿ for all z in U − {a}.

• The point a is an essential singularity of f if it is neither a removable singularity nor a pole.

• A branch point of f is one requiring a more verbose definition; see the article of that title.

From the point of view of dynamics

A finite-time 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 finite-time singularities include Euler's disk and the Painlevé paradox .