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

(* Specifications: *)
Lemma continuous_func_preserves_net_limits:
forall {I:DirectedSet} (x:Net I X) (x0:point_set X),
net_limit x x0 -> continuous_at f x0 ->
net_limit (fun i:DS_set I => f (x i)) (f x0).
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.