Definition continuous_at(f : Rat -> Rat -> Prop)(c : Rat) :=
  forall v, f c v ->
    rat_limit f c v.

Lemma rat_inf_limit_mono : forall (f : nat -> Rat -> Prop) (g : nat -> nat)(v : Rat), rat_inf_limit f v -> (forall n1 n2, n1 <= n2 -> g n1 <= g n2)%nat -> (forall y, exists x, g x = y) -> rat_inf_limit (fun n => f (g n)) v.