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