Removable Discontinuity/Examples

Examples of Removable Discontinuities

Example 1

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

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

Then $f$ has a removable discontinuity at $x = 1$.

In this case the removable discontinuity may be removed by defining $\map f 1$ to equal $2$.

Example 2

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

$\forall x \in \R: \map f x = \begin {cases} x \map \sin {\dfrac 1 x} & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Then $f$ has a removable discontinuity at $x = 0$.

In this case the removable discontinuity may be removed by redefining $\map f 0$ to equal $0$.