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. 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. 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. 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. 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 $\overline{A}$

1. 1.

   Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by

   $$f(x)=\begin{cases}1&\text{when }x\neq 0,\\ 0&\text{when }x=0.\end{cases}$$

   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. 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. 3.

   The signum function $\mathop{\mathrm{sign}}\colon\mathbb{R}\to\mathbb{R}$ is defined as

   $$\mathop{\mathrm{sign}}(x)=\begin{cases}-1&\text{when }x<0,\\ 0&\text{when }x=0,\text{ and}\\ 1&\text{when }x>0.\end{cases}$$

   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. 4.

   The function $f:\mathbb{R}\to\mathbb{R}$ given by

   $$f(x)=\begin{cases}1&\text{when }x=0,\\ \sin(1/x)&\text{when }x\neq 0\end{cases}$$

   has an essential discontinuity at $x=0$. See [2] for details.

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.

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.