Equivalence of Definitions of Removable Discontinuity of Real Function

Theorem
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 $a \in A$.

Lemma
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 $a \in A$.

For any $b \in \R$, define the function $f_b$ by:
 * $\map {f_b} x = \begin {cases} \map f x &: x \ne a \\ b &: x = a \end {cases}$

Then $\map {f_b} x = \map f x$ for every $x \ne a$.

Definition 1 implies Definition 2
Suppose the limit $\ds \lim_{x \mathop \to a} \map f x$ exists.

Let $\ds \lim_{x \mathop \to a} \map f x = b$.

Then:

By definition of continuity, $f_b$ is continuous at $a$.

Definition 2 implies Definition 1
Suppose there exists $b \in \R$ such that $f_b$ is continuous at $a$.

By definition of continuity:


 * $\ds \lim_{x \mathop \to a} \map {f_b} x = \map {f_b} a = b $

Then by the lemma:


 * $\ds \lim_{x \mathop \to a} \map f x = \lim_{x \mathop \to a} \map {f_b} x = b$

It follows that the limit $\ds \lim_{x \mathop \to a} \map f x$ exists.