Definition:Removable Discontinuity of Real Function

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 $a\in A$.

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

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 $a$.