Definition:Discontinuity (Real Analysis)/Removable

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 $c \in X$.

The point $c$ is a removable discontinuity of $f$ if and only if the limit $\ds \lim_{x \mathop \to c} \map f x$ exists.

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 $c$.

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 $2$.

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

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

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 $0$.

Results about removable discontinuities in the context of Real Analysis can be found here.