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.