Definition:Removable Discontinuity

Definition

Real Function

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 $c \in X$.

Definition 1

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.

Definition 2

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 $c$.

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Also see

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

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): removable