Let E be a topological space and F a uniform space. We introduce a new topology (in fact a uniform structure) called the V-congergence on the space of applications from E to F such that C(E,F) is closed for this topology and the restriction of this topology to C(E,F) is equivalent to pointwise convergence. In other words this topology is the coarsest preserving continuity. We give a criterion of convergence for this topology not involving the limit. Among properties preserved are mesurability and alpha-borelianity for a countable ordinal alpha.