= 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 $[\alpha]\Longleftrightarrow[\beta]$.
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 $[\beta]\Longleftrightarrow[\gamma]$. Hence \b[all three conditions are equivalent].
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