PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] discontinuous (Definition)

Definition

Suppose $ A$ is an open set in $ \mathbb{R}$ (say an interval $ A=(a,b)$, or $ A=\mathbb{R}$), and $ f:A\to \mathbb{R}$ is a function. Then $ f$ is discontinuous at $ x\in A$, 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 $ \lim_{z\to x} f(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+)\neq f(x)$, or $ f(x-)\neq f(x)$.

If $ f$ is discontinuous at $ x\in\overline{A}$, the closure of $ A$, we can then distinguish four types of different discontinuities as follows [1,2]:

  1. If $ f(x+)=f(x-)$, but $ f(x)\neq f(x\pm)$, then $ x$ is called a removable discontinuity of $ f$. If we modify the value of $ f$ at $ x$ to $ f(x)=f(x\pm)$, then $ f$ will become continuous at $ x$. This is clear since the modified $ f$ (call it $ f_0$) satisfies $ f_0(x) = f_0(x+)=f_0(x-).$
  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\pm)$, then this modification renders $ f$ continuous at $ x$.
  3. If $ f(x-)\neq 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. 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 $ \overline{A}$

Examples

  1. Consider the function $ f:\mathbb{R}\to \mathbb{R}$ given by
    $\displaystyle f(x)=\begin{cases} 1 & \text{when }x\neq 0, \ 0 & \text{when }x=0. \end{cases}$
    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. Let us consider the function defined by the formula
    $\displaystyle f(x) = \frac{\sin x }{x} $
    where $ x$ is a nonzero real number. When $ x=0$, the formula is undefined, so $ f$ is only determined for $ x\neq 0$. Let us show that this point is a removable discontinuity. Indeed, it is easy to see that $ f$ is continuous for all $ x\neq 0$, and using L'Hôpital's rule 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 function on the whole complex plane.
  3. The signum function $ \mathop{\mathrm{sign}}\colon\mathbb{R}\to \mathbb{R}$ is defined as
    $\displaystyle \mathop{\mathrm{sign}}(x) =\begin{cases} -1 & \text{when }x<0, \ 0 & \text{when } x=0, \text{ and}\ 1 & \text{when } x>0. \end{cases}$
    Since $ \mathop{\mathrm{sign}}(0+)=1$, $ \mathop{\mathrm{sign}}(0)=0$, and since $ \mathop{\mathrm{sign}}(0-)=-1$, it follows that $ \mathop{\mathrm{sign}}$ has a jump discontinuity at $ x=0$ with jump $ \mathop{\mathrm{sign}}(0+)-\mathop{\mathrm{sign}}(0-)=2$.
  4. The function $ f:\mathbb{R}\to\mathbb{R}$ given by
    $\displaystyle f(x) =\begin{cases} 1 & \text{when }x= 0, \ \sin(1/x) & \text{when } x\neq 0 \end{cases}$
    has an essential discontinuity at $ x=0$. See [2] for details.

General Definition

Let $ X,Y$ be topological spaces, and let $ f$ be a mapping $ f:X\to Y$. Then $ f$ is discontinuous at $ x\in X$, if $ f$ is not continuous at $ x$.

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

Notes

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.

References

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.pdf.



"discontinuous" is owned by mathwizard. [ full author list (3) | owner history (2) ]
(view preamble)

View style:

Also defines:  removable discontinuity, saltus, jump, jump discontinuity, discontinuity of the second kind, discontinuity of the first kind, essential discontinuity

This object's parent.

Attachments:
example of jump discontinuity (Example) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: mapping, topological spaces, signum function, complex plane, analytic function, real number, continuous, number, modification, clear, closure, one-sided limits, properties, points, boundary, continuous at, function, interval, open set
There are 31 references to this entry.

This is version 11 of discontinuous, born on 2003-07-14, modified 2006-10-30.
Object id is 4447, canonical name is Discontinuous.
Accessed 21640 times total.

Classification:
AMS MSC: 54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)