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

If continuity holds everywhere, it holds at $a$. Conversely, if it holds at $a$, it holds at every other point automatically: the singleton neighborhood from (ii) maps entirely to the single value at that point. Thus continuity everywhere in $d_a$ is equivalent to continuity at $a$ in $d_a$.