  The \emph{Continuity Problem} is the question whether effective operators are
continuous, where an effective operator $F$ is a function on a space of
constructively given objects $x$, defined by mapping construction instructions
for $x$ to instructions for $F(x)$ in a computable way. In the present paper
the problem is dealt with in a bi-topological setting. To this end the
topological setting developed by the author \cite{sp} is extended to the
bi-topological case. Under very natural conditions it is shown that an
effective operator $F$ between bi-topological spaces $\TTT = (T, \tau, \sigma)$
and $\TTT' = (T', \tau', \sigma')$ is (effectively) continuous, if $\tau'$ is
(effectively) regular with respect to $\sigma'$. A central requirement on
$\TTT'$ is that bases of the neighbourhood filters of the points in $T'$ can
computably be enumerated in a uniform way, not only with respect to topology
$\tau'$, but also with respect to $\sigma'$. As follows from an example by
Friedberg, the last condition is indispensable. Conversely, it is proved that
(effectively) bi-continuous operators are effective. A prominent example of
bi-topological spaces are quasi-metric spaces. Under a very reasonable
computability requirement on the quasi-metric it is shown that all effectivity
assumptions made in the general results are satisfied in the quasi-metric case.

Logic;General Topology