Suppose A is an open set in (say an interval A=(a,b), or A=), and f:A is a function. Then f is discontinuousMathworldPlanetmath at xA, if f is not continuous at x. One also says that f is discontinuous at all boundary points of A.

We know that f is continuous at x if and only if limzxf(z)=f(x). Thus, from the properties of the one-sided limits, which we denote by f(x+) and f(x-), it follows that f is discontinuous at x if and only if f(x+)f(x), or f(x-)f(x).

If f is discontinuous at xA¯, the closurePlanetmathPlanetmath of A, we can then distinguish four types of different discontinuities as follows [1, 2]:

  1. 1.

    If f(x+)=f(x-), but f(x)f(x±), then x is called a removable discontinuity of f. If we modify the value of f at x to f(x)=f(x±), then f will become continuous at x. This is clear since the modified f (call it f0) satisfies f0(x)=f0(x+)=f0(x-).

  2. 2.

    If f(x+)=f(x-), but x is not in A (so f(x) is not defined), then x is also called a removable discontinuity. If we assign f(x)=f(x±), then this modification renders f continuous at x.

  3. 3.

    If f(x-)f(x+), then f has a jump discontinuity at x Then the number f(x+)-f(x-) is then called the jump, or saltus, of f at x.

  4. 4.

    If either (or both) of f(x+) or f(x-) does not exist, then f has an essential discontinuity at x (or a discontinuity of the second kind).

Note that f may be continuous (continuous in all points in A), but still have discontinuities in A¯


  1. 1.

    Consider the function f: given by

    f(x)={1when x0,0when x=0.

    Since f(0-)=1, f(0)=0, and f(0+)=1, it follows that f has a removable discontinuity at x=0. If we modify f(0) so that f(0)=1, then f becomes the continuous function f(x)=1.

  2. 2.

    Let us consider the function defined by the formula


    where x is a nonzero real number. When x=0, the formula is undefined, so f is only determined for x0. Let us show that this point is a removable discontinuity. Indeed, it is easy to see that f is continuous for all x0, and using L’Hôpital’s rule (http://planetmath.org/LHpitalsRule) we have f(0+)=f(0-)=1. Thus, if we assign f(0)=1, then f becomes a continuous function defined for all real x. In fact, f can be made into an analytic functionMathworldPlanetmath on the whole complex plane.

  3. 3.

    The signum function sign: is defined as

    sign(x)={-1when x<0,0when x=0, and1when x>0.

    Since sign(0+)=1, sign(0)=0, and since sign(0-)=-1, it follows that sign has a jump discontinuity at x=0 with jump sign(0+)-sign(0-)=2.

  4. 4.

    The function f: given by

    f(x)={1when x=0,sin(1/x)when x0

    has an essential discontinuity at x=0. See [2] for details.

General Definition

Let X,Y be topological spacesMathworldPlanetmath, and let f be a mapping f:XY. Then f is discontinuous at xX, if f is not continuous at (http://planetmath.org/Continuous) x.

In this generality, one generally does not classify discontinuities quite so closely, since they can have quite complicated behaviour.


A jump discontinuity is also called a simple discontinuity, or a discontinuity of the first kind. An essential discontinuity is also called a discontinuity of the second kind.


  • 1 R.F. Hoskins, Generalised functions, Ellis Horwood Series: Mathematics and its applications, John Wiley & Sons, 1979.
  • 2 P. B. Laval, http://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdfhttp://science.kennesaw.edu/ plaval/spring2003/m4400_02/Math4400/contwork.pdf.
Title discontinuous
Canonical name Discontinuous
Date of creation 2013-03-22 13:45:01
Last modified on 2013-03-22 13:45:01
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 14
Author mathwizard (128)
Entry type Definition
Classification msc 26A15
Classification msc 54C05
Defines removable discontinuity
Defines saltus
Defines jump
Defines jump discontinuity
Defines discontinuity of the second kind
Defines discontinuity of the first kind
Defines essential discontinuity