Mathematical Singularity
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A Mathematical Singularity is a mathematical 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: Tangent, Mathematics, Set (Mathematics), Well-Behaved, Derivative, Singularity Theory, Geometric, Function (Mathematics), Real Line, Absolute Value, Differentiable Function, Technological Singularity.
References
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/singularity_(mathematics) Retrieved:2017-6-12.
- 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 : [math]\displaystyle{ f(x)=\frac{1}{x} }[/math] 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.
The algebraic set defined by [math]\displaystyle{ \{(x,y):|x|=|y|\} }[/math] in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.
- 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.