Section Nets_and_continuity.
Variable X Y:TopologicalSpace.
Variable f:point_set X -> point_set Y.

Lemma func_preserving_net_limits_is_continuous:
forall x0:point_set X,
(forall (I:DirectedSet) (x:Net I X),
net_limit x x0 -> net_limit (fun i:DS_set I => f (x i)) (f x0))
-> continuous_at f x0.