Equivalence of Definitions of Removable Discontinuity of Real Function/Lemma

Lemma for Equivalence of Definitions of Removable Discontinuity of Real Function

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 $x \ne a$.

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

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

Proof

This can be regarded as a corollary of the Squeeze Theorem for Functions.

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

We have:

$\forall x \ne a: \map f x \le \map g x \le \map f x$

Hence by the Squeeze Theorem for Functions:

$\ds \lim_{x \mathop \to a} \map g x = L$

$\blacksquare$