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Mathematical singularity

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)=\frac{1}{x}

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.

Real analysis
In [real analysis] singularities are also called [discontinuities] . There are three kinds: type I , which has two sub-types, 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:

:f(c^ ) = \lim_{x \to c}f(x), constrained by x > c\ and

:f(c^-) = \lim_{x \to c}f(x), constrained by x < c\ .

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