# Definition:Discontinuity (Real Analysis)

## Definition

A **discontinuity** is an element $D$ in the domain of a real function $f$ at which $f$ is discontinuous.

### Jump Discontinuity

Let $X$ be an open subset of $\R$.

Let $f: X \to Y$ be a real function.

Let $f$ be discontinuous at some point $c \in X$.

Then $c$ is called a **jump discontinuity** of $f$ if and only if:

- $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ exist and are not equal

where $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively.

Note that $\map f c$ may equal either of these limits, or neither, or may not even be defined.

### Removable Discontinuity

The point $c$ is a **removable discontinuity** of $f$ if and only if there exists $b \in \R$ such that the function $f_b$ defined by:

- $\map {f_b} x = \begin {cases} \map f x &: x \ne c \\ b &: x = c \end {cases}$

is continuous at $c$.

### Non-Removable Discontinuity

The point $a$ is a **non-removable discontinuity** of $f$ if and only if it is not a **removable discontinuity**.

### Discontinuity of the First Kind

Let $X$ be an open subset of $\R$.

Let $f: X \to Y$ be a real function.

Let $f$ be discontinuous at some point $c \in X$.

$c$ is known as a **discontinuity of the first kind** of $f$ if and only if either:

- $c$ is a
**jump discontinuity**

or:

- $c$ is a
**removable discontinuity**.

### Simple Discontinuity

This is the same thing as a **discontinuity of the first kind**.

### Finite Discontinuity

Let $X \subseteq \R$ be a subset of the real numbers.

Let $f: X \to \R$ be a real function.

Let $f$ be discontinuous at $c \in X$.

$f$ is a **finite discontinuity** on $f$ if and only if $\map f c$ is finite.

### Infinite Discontinuity

Let $X \subseteq \R$ be a subset of the real numbers.

Let $f: X \to \R$ be a real function.

Let $f$ be discontinuous at $c \in X$.

$f$ is an **infinite discontinuity** on $f$ if and only if $\size {\map f x}$ becomes arbitrarily large as $\size {x - c}$ becomes arbitrarily small.

## Examples

### Example 1

Let $f: \R \to \R$ be the real function defined as:

- $\forall x \in \R: \map f x = \dfrac 1 {1 - x}$

Then $f$ has a **discontinuity** at $x = 1$, as $f$ is not defined there.

### Example 2

Let $f: \R \to \R$ be the real function defined as:

- $\forall x \in \R: \map f x = \begin {cases} \dfrac 1 k & : 0 \le x \le k \\ 0 & : \text {otherwise} \end {cases}$

Then $f$ has a **discontinuity** at $x = 0$ and $x = k$

### Example 3

Let $f: \R \to \R$ be the real function defined as:

- $\forall x \in \R: \map f x = \dfrac 1 {x^2 - 4}$

Then $f$ has a **discontinuity** at $x = -2$ and $x = 2$.

These are infinite discontinuities.

## Also see

- Results about
**discontinuities**in the context of**real analysis**can be found**here**.

## Linguistic Note

The word **discontinuity**, as well as meaning an element of the domain at which a mapping is discontinuous, can also be used in the abstract sense as meaning **the property of being discontinuous**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**discontinuity**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**discontinuity**