Mathematical singularity

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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

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 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.


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:

The limit f(c ) is called the left-handed limit, and f(c + ) is called the right-handed 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.

  • A point of continuity, which is not a singularity, is a value of c for which f(c ) = f(c) = f(c + ), as one usually expects. All the values must be finite.
  • A type I discontinuity occurs when both f(c ) and f(c + ) exist and are finite, but one of three conditions also apply: f(c^-) \neq f(c^+); f(c) does not exist for that value of x; or f(c) does not match the value that the two limits tend towards. Two subtypes occur:
    • A jump discontinuity occurs when f(c^-) \neq f(c^+), regardless of whether f(c) exists, and regardless of what value it might have if it does exist.
    • A removable discontinuity occurs when f(c ) = f(c + ), but either the value of f(c) does not match the limits, or the function does not exist at the point x = c .
  • A type II discontinuity occurs when either f(c ) or f(c + ) does not exist (possibly both). This has two subtypes, which are usually not considered separately:
    • An infinite discontinuity is the special case when either the left hand or right hand limit does not exist specifically because it is infinite, and the other limit is either also infinite or is some well defined finite number.
    • An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits f(c ) or f(c + ) does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if legal answers are extended to include \pm\infty.

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