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[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.



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

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example of jump discontinuity (Example) by pahio
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Cross-references: mapping, topological spaces, signum function, complex plane, analytic function, real number, continuous, modification, clear, closure, one-sided limits, properties, points, boundary, continuous at, function, interval, open set
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This is version 11 of discontinuous, born on 2003-07-14, modified 2006-10-30.
Object id is 4447, canonical name is Discontinuous.
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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 )
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