Definition:Removable Discontinuity
Jump to navigation
Jump to search
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 $a$.