  We axiomatize and generalize Markov's approach to the continuity problem for
Type 1 computable functions, i.e. the problem of finding sufficient conditions
on a computable topological space to obtain a theorem of the form "computable
functions are (effectively) continuous". In a computable topological space, a
point $x$ is called effectively adherent to a set $A$ if there is an algorithm
that on input a neighborhood of $x$ produces a point of $A$ in that
neighborhood. We say that a space $X$ satisfies a Markov condition if, whenever
a point $x$ of $X$ is effectively adherent to a subset $A$ of $X$, the
singleton $\{x\}$ is not a semi-decidable subset of $A\cup\{x\}$. We show that
this condition prevents functions whose domain is $X$ from having effective
discontinuities, provided that their codomain is a space where points have
neighborhood bases of co-semi-decidable sets. We then show that results that
forbid effective discontinuities can be turned into (abstract) continuity
results on spaces where the closure and effective closure of semi-decidable
sets naturally agree -this happens for instance on spaces which admit a dense
and computable sequence. This work is motivated by the study of the space of
marked groups, for which the author has shown that most known continuity
results (of Ceitin, Moschovakis, Spreen) do not apply.
