discontinuous

Definition

Suppose $A$ is an open set in $ℝ$ (say an interval $A=(a,b)$, or $A=ℝ$), and $f:A\to ℝ$ 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 $f(x+)\ne f(x)$, or $f(x-)\ne f(x)$.

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 $f(x)\ne f(x\pm )$, 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 $f(x-)\ne f(x+)$, 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 $f:ℝ\to ℝ$ given by

   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

   $$f(x)=\frac{\sin x}{x}$$

   where $x$ is a nonzero real number. When $x=0$, the formula is undefined, so $f$ is only determined for $x\ne 0$. Let us show that this point is a removable discontinuity. Indeed, it is easy to see that $f$ is continuous for all $x\ne 0$, 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 function on the whole complex plane.

3. The signum function $sign:ℝ\to ℝ$ is defined as

   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. The function $f:ℝ\to ℝ$ given by

   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 (http://planetmath.org/Continuous) $x$.

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.

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