Definition:Removable Discontinuity of Real Function
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Definition
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 $\displaystyle \lim_{x \to a}f(x)$ 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:
- $f_b(x) = \begin{cases}f(x) &: x \neq a\\ b &: x = a\end{cases}$
is continuous at $a$.
Also see
- Equivalence of Definitions of Removable Discontinuity of Real Function
- Definition:Nonremovable Discontinuity
- Definition:Jump Discontinuity
- Definition:Discontinuity of the First Kind
Sources
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 1.4$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: discontinuity