Definition continuous_on (D : T -> Prop) (f : T -> U) :=
  forall x, D x -> filterlim f (within D (locally x)) (locally (f x)).Definition continuous (f : T -> U) (x : T) :=
  filterlim f (locally x) (locally (f x)).

Lemma is_lim_comp_continuous (f g : R -> R) (x : Rbar) (l : R) : is_lim f x l -> continuous g l -> is_lim (fun x => g (f x)) x (g l).