Section Continuity.
Context {T U : UniformSpace}.
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)).
End Continuity.

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).