phi(x) is continuous for x = xi if, given delta, we can choose
epsilon(delta) so that |phi(x)-phi(xi)| < delta if
0 <= |x-xi| <= epsilon(delta).
with the remark that he would define continuity only for points such that
an entire neighbourhood of them would be in the domain of phi . So his
definition of continuity of phi at a point x of its domain A has really
two clauses
(1) A contains an open interval containing x
(2) the epsilon-delta condition above.
And if one of these conditions is not satisfied, then he calls phi
discontinuous at x. Then indeed the restriction of Dirichlet's function to
the rationals is discontinuous.
It may be difficult to say whether Hardy's clause (1) generally was
included into the definitions of continuity in the case of functions with
arbitrary domains. Hobson, in his "Theory of functions of a real variable",
another Cambridge book, third edition 1924, did not do so, rather he
defined f with domain A to be continuous at x in A if
for every e>0 : there exists d>0 : for every d' with |d'|<d :
if x+d' in A then |f(x) - f(x+d')| < e .
Hardy certainly did not use Hobson's definition, and in order to say
whether there was a 'standard' definition at that time, it might be
necessary to look at other of its authors how, if they treated them at
all, they proceeded in the case of arbitrary domains.
The odd thing, of course, is that Hardy, by stating (1) as a preliminary
to his definition, may at first have supposed that he only wanted to
restrict himself to a simpler case. But for negated continuity,
discontinuity, he then abandoned (1) as preliminary (which would have
led to "(1) and not (2)" , but rather used "not (1) or not (2)".
WF