Characterization of Continuity

If $f : A \to R$ where $A \subseteq R$ and $c \in A$ , then the following are equivalent:

$f$ is continuous at $c$ .
$\forall ε > 0 \exists δ > 0 \forall x \in A (x \in A \cap V_{ε} (c) \to f (x) \in V_{ε} (f (c)))$
$\forall (x_{n}) \subseteq A$ where $x_{n} \to c$ , then $f (x_{n}) \to f (c)$ .
If specifically $c$ is a Limit Point of $A$ , then each of these is equivalent to:

lim_{x \to c} f (x) = f (c)

Proof

Notice the first one is just a rewording using Epsilon-Neighborhood notation. Thus $(1) ⟺ (2)$ .

#todo/school : Finish this proof 📅 2024-11-08 ✅ 2024-11-06

( $(3) ⟺ (1)$ ): Use an argument similar to the Sequential Criterion For Functional Limits, with some modifications for when $x_{n} = c$ .

( $(4) ⟺ (1)$ ): Consider the Functional Limit definition. Just observe the case that $x = c$ (which isn't included in the definition for Functional Limits) leads to the requirement that $f (c) \in V_{ε} (f (c))$ , which is trivially true.

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As a bit of a corollary, we can get a characterization for discontinuity.

Criterion for Discontinuity

Let $f : A \to R$ and let $c \in A$ be a limit point of $A$ . If there exists a sequence $(x_{n}) \subseteq A$ where $(x_{n}) \to c$ but such that $f (x_{n}) ↛ f (c)$ , we may conclude that $f$ is not continuous at $c$ .