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 continuous_comp_2 {U V W X : UniformSpace}
(f : U -> V) (g : U -> W) (h : V -> W -> X) (x : U) :
continuous f x -> continuous g x
-> continuous (fun (x : V * W) => h (fst x) (snd x)) (f x,g x)
-> continuous (fun x => h (f x) (g x)) x.