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_ext : forall (D : T -> Prop) (f g : T -> U), (forall x, D x -> f x = g x) -> continuous_on D f -> continuous_on D g.