Proof using.
intros f P Q eP eQ QP.
apply always_map_conv; apply eventually_map_conv; trivial.
Qed.

Lemma continuously_map :
forall (f: A->B) (P: infseq A->Prop) (Q: infseq B->Prop),
(forall s, P s -> Q (map f s)) ->
forall (s: infseq A), continuously P s -> continuously Q (map f s).