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 $f, g: A \to \R$ be real functions.

Let $a \in A$.

Suppose $\map f x = \map g x$ for every $x \neq a$.

Suppose the limit $\displaystyle \lim_{x \to a} \map f x$ exists.

Then the limit $\displaystyle \lim_{x \to a} \map g x$ exists and is equal to $\displaystyle \lim_{x \to a} \map f x$.

Proof of Lemma
This Lemma can be regarded as a corollary of Squeeze Theorem.

Let $\displaystyle \lim_{x \to a} \map f x = L$.

Since we have $\map f x \le \map g x \le \map f x \; \forall x \neq a$, We have $\displaystyle \lim_{x \to a} \map g x = L$ by Squeeze Theorem.

Proof of 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 $a\in A$.

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 $x \neq a$.

Definition 1 implies Definition 2
Suppose the limit $\displaystyle \lim_{x \to a} \map f x$ exists. Let $\displaystyle \lim_{x \to a} \map f x = b$. 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:

$\displaystyle \lim_{x \to a} \map {f_b} x = \map {f_b} a = b $.

Then $\displaystyle \lim_{x \to a} \map f x = \lim_{x \to a} \map {f_b} x = b$, by Lemma.

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