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 {U V W : UniformSpace} (f : U -> V) (g : V -> W) (x : U) :
continuous f x -> continuous g (f x)
-> continuous (fun x => g (f x)) x.