Definition:Removable Discontinuity
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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