Equivalence of Definitions of Metric Space Continuity at Point

Theorem
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Let $a \in A_1$ be a point in $A_1$.

$\epsilon$-$\delta$ Definition iff Definition by Limits
This is proved in Metric Space Continuity by Epsilon-Delta.

$\epsilon$-$\delta$ Definition iff $\epsilon$-Ball Definition
This is proved in Metric Space Continuity by Open Ball.

$\epsilon$-Ball Definition iff Definition by Neighborhood
This is proved in Metric Space Continuity by Neighborhood.