The same equality of balls gives the reverse implication: <continuity> at $a$ in $d$ implies <continuity> at $a$ in $d_a$. The latter implies <continuity> everywhere by the first part. These implications <complete> the equivalence of all three printed conditions.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E3 / b / iii / γ / Solution

 Solution (source code)