 Solution

The singleton calculation shows that every map out of $(X,d_a)$ is continuous at each point other than $a$: use the domain ball consisting only of that point. Thus global continuity in this metric is equivalent to continuity at $a$, proving $[\alpha ]⟺[\beta ]$.

For every $x$, including $x=a$, one has $d_a(a,x)=d(a,x)$. Therefore the open balls centred at $a$ are identical for $d_a$ and $d$. The epsilon-delta definitions of continuity at $a$ are identical, giving $[\beta ]⟺[\gamma ]$. Hence all three conditions are equivalent.