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_id {U : UniformSpace} (x : U) :
continuous (fun y => y) x.