Equivalence of Definitions of Metric Space Continuity at Point

Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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 $A_1$.

Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:

$\epsilon$-$\delta$ 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.