Mathematical Singularity - Real Analysis

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 is a function of a real argument, and for any value of its argument, say, the symbols and are defined by:

, constrained by and

, constrained by .

The limit is called the left-handed limit, and is called the right-handed limit. The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where .

There are some functions for which these limits do not exist at all. For example the function

does not tend towards anything as approaches . The limits in this case are not infinite, but rather undefined: there is no value that 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 for which, as one usually expects. All the values must be finite.
• A type I discontinuity occurs when both and exist and are finite, but one of three conditions also apply: ; does not exist for that value of ; or does not match the value that the two limits tend towards. Two subtypes occur:
  • A jump discontinuity occurs when, regardless of whether exists, and regardless of what value it might have if it does exist.
  • A removable discontinuity occurs when, but either the value of does not match the limits, or the function does not exist at the point .
• A type II discontinuity occurs when either or 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 or 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 .

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.