# Dictionary Definition

discontinuous adj

1 of a function or curve; possessing one or more
discontinuities [ant: continuous]

2 not continuing without interruption in time or
space; "discontinuous applause"; "the landscape was a discontinuous
mosaic of fields and forest areas"; "he received a somewhat
haphazard and discontinuous schooling" [syn: noncontinuous] [ant:
continuous]

# User Contributed Dictionary

## English

### Adjective

discontinuous- having breaks or interruptions; intermittent
- having at least one discontinuity

#### Translations

*Swedish: diskontinuerlig# Extensive Definition

- Jump point redirects here. For the book by Tom Hayes, see Jump Point.

This article describes the classification of
discontinuities in the simplest case of functions of a single
real
variable taking real values.

## Classification of discontinuities

Consider a function f of real variable x with
real values defined in a neighborhood of a point x_0. Then three
situations are possible:

1. The one-sided
limit from the negative direction

- L^=\lim_ f(x)

- L^=\lim_ f(x)

2. The limits L^ and L^ exist and are finite, but
not equal. Then, x0 is called a jump discontinuity or step
discontinuity.

3. One or both of the limits L^ and L^ does not
exist or is infinite. Then, x0 is called an essential
discontinuity.

The term removable discontinuity is sometimes
(improperly) used for cases in which the limits in both directions
exist and are equal, while the function is undefined
at the point x_0. This use is improper because continuity
and discontinuity of a function are concepts defined only for
points in the function's domain.

## Examples

1. Consider the function- f(x)=\beginx^2 & \mbox x1\end

2. Consider the function

- f(x)=\beginx^2 & \mbox x1\end

3. Consider the function

- f(x)=\begin\sin\frac & \mbox x1\end

## The set of discontinuities of a function

The set of points at which a function is
continuous is always a Gδ set. The
set of discontinuities is an Fσ set.

Thomae's
function is discontinuous at every rational point, but
continuous at every irrational point.

The indicator
function of the rationals, also known as the Dirichlet
function, is discontinuous
everywhere.

## See also

## Notes

## References

## External links

discontinuous in German: Sprungstetigkeit

discontinuous in Spanish: Clasificación de
discontinuidades

discontinuous in Italian: Punto di
discontinuità

discontinuous in Hebrew: נקודת אי רציפות

discontinuous in Korean: 불연속성의 분류

discontinuous in Dutch: Discontinuïteit

discontinuous in Swedish: Diskontinuitet

discontinuous in Chinese: 间断点

# Synonyms, Antonyms and Related Words

aimless, aloof, amorphous, apart, asunder, bipartite, broken, broken off, capricious, careening, casual, catchy, chopped-off, choppy, decousu, desultory, detached, deviative, dichotomous, disarticulated, disconnected, discontinued, discrete, disjointed, disjunct, disjunctive, disordered, dispersed, disproportionate,
distinct, divergent, eccentric, episodic, erratic, fitful, flickering, fluctuating, formless, frivolous, gapped, gratuitous, guttering, halting, haphazard, herky-jerky,
heteroclite,
hit-or-miss, immethodical, in two,
inchoate, incoherent, inconsistent, inconstant, indiscriminate, insular, intermittent, intermitting, interrupted, irregular, jagged, jerky, lurching, meaningless, misshapen, muddled, nonadherent, nonadhesive, noncoherent, noncohesive, noncontinuous, nonlinear, nonsequential, nonserial, nonsymmetrical, nonsystematic, nonuniform, open, orderless, parenthetic, partitioned, patchy, planless, promiscuous, rambling, random, rough, scrappy, senseless, separate, shapeless, snatchy, spasmatic, spasmic, spasmodic, spastic, sporadic, spotty, staggering, straggling, straggly, suspended, systemless, tenuous, unadhesive, unarranged, unassociated, unattached, unattended, uncertain, unclassified, uncoherent, uncohesive, unconnected, undirected, unequal, uneven, ungraded, unjoined, unmethodical, unmetrical, unordered, unorganized, unregular, unrhythmical, unsettled, unsorted, unsteady, unsuccessive, unsymmetrical, unsystematic, untenacious, ununiform, vague, variable, veering, wandering, wavering, wobbling, wobbly