Category:Equivalence of Definitions of Removable Discontinuity of Real Function

This category contains pages concerning Equivalence of Definitions of Removable Discontinuity of 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 $a \in A$.

The following definitions of the concept of removable discontinuity are equivalent:

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 $c$.