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_plus (f g : U -> V) (x : U) : continuous f x -> continuous g x -> continuous (fun x : U => plus (f x) (g x)) x.