Definition:Removable Discontinuity

Definition

Real Function

Let $A \subseteq \R$ be a subset of the real numbers.

Let $f: A \to \R$ be a real function.

Let $f$ be discontinuous at $a\in A$.

Definition 1

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

Definition 2

The point $a$ is a removable discontinuity of $f$ if and only if there exists $b \in \R$ such that the function $f_b$ defined by:

$\map {f_b} x = \begin {cases} \map f x &: x \ne a \\ b &: x = a \end {cases}$

is continuous at $a$.

Also see

• Definition:Isolated Singularity
• Definition:Nonremovable Discontinuity
• Definition:Jump Discontinuity
• Definition:Discontinuity of the First Kind