Definition:Discontinuity (Real Analysis)/Finite

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

Examples

Example 1

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

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

Then $f$ has a finite discontinuity at $x = 0$ which is non-removable.

Also see

• Definition:Infinite Discontinuity

• Results about finite 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): finite 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): finite discontinuity