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)).

Lemma continuous_on_forall :
forall (D : T -> Prop) (f : T -> U),
(forall x, D x -> continuous f x) ->
continuous_on D f.