Definition

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).

Examples

1. Consider the function $f:\sR\to \sR$ 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 $x\neq 0$ . 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 $\signum\colon\sR\to \sR$ is defined as
   
   Since $\signum(0+)=1$ , $\signum(0)=0$ , and since $\signum(0-)=-1$ , it follows that $\signum$ has a jump discontinuity at $x=0$ with jump $\signum(0+)-\signum(0-)=2$ .
4. The function $f:\sR\to\sR$ given by
   
   has an essential discontinuity at $x=0$ . See [2] for details.

General Definition

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

Notes

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.