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 continuousat $x$. One also says that $f$ is discontinuous at all boundary points of $A$.

We know that $f$ is continuous at $x$ if and onlyif ${lim}_{z\to x}f(z)=f(x)$. Thus, from the properties of theone-sided limits, which we denote by $f(x+)$ and $f(x-)$, it followsthat $f$ 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 ofdifferent 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 $f(x)$ isnot 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 thecontinuous 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 isa removable discontinuity. Indeed, it is easyto see that $f$ is continuous for all $x\ne 0$, and usingL’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 functiondefined for all real $x$. In fact, $f$ can be made into an analyticfunction 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 notcontinuous 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 adiscontinuity of the second kind.