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 unifcont_normed_1d (f : R -> V) a b : (forall x, a <= x <= b -> continuous f x) -> forall eps : posreal, {delta : posreal | forall x y, a <= x <= b -> a <= y <= b -> ball x delta y -> ball_norm (f x) eps (f y)}.