Definition:Discontinuity (Real Analysis)/Infinite

Definition

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.

Also known as

An infinite discontinuity is also known as a singularity.

However, this has other similar yet different uses, so it is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Examples

Example 1

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

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

Then $f$ has an infinite discontinuity at $x = 0$.

Also see

• Definition:Finite Discontinuity

• Results about infinite discontinuities can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discontinuity
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinite discontinuity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discontinuity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinite discontinuity