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)).

(* Specifications: *)
Lemma continuous_continuous_on : forall (D : T -> Prop) (f : T -> U) (x : T), locally x D -> continuous_on D f -> continuous f x.
Lemma continuous_on_subset : forall (D E : T -> Prop) (f : T -> U), (forall x, E x -> D x) -> continuous_on D f -> continuous_on E f.
Lemma continuous_on_forall : forall (D : T -> Prop) (f : T -> U), (forall x, D x -> continuous f x) -> continuous_on D f.
Lemma continuous_ext_loc (f g : T -> U) (x : T) : locally x (fun y : T => g y = f y) -> continuous g x -> continuous f x.
Lemma continuous_ext : forall (f g : T -> U) (x : T), (forall x, f x = g x) -> continuous f x -> continuous g 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.
Lemma continuous_comp {U V W : UniformSpace} (f : U -> V) (g : V -> W) (x : U) : continuous f x -> continuous g (f x) -> continuous (fun x => g (f x)) x.
Lemma continuous_comp_2 {U V W X : UniformSpace} (f : U -> V) (g : U -> W) (h : V -> W -> X) (x : U) : continuous f x -> continuous g x -> continuous (fun (x : V * W) => h (fst x) (snd x)) (f x,g x) -> continuous (fun x => h (f x) (g x)) x.
Lemma is_lim_comp_continuous (f g : R -> R) (x : Rbar) (l : R) : is_lim f x l -> continuous g l -> is_lim (fun x => g (f x)) x (g l).
Lemma continuous_fst {U V : UniformSpace} (x : U) (y : V) : continuous (fst (B:=V)) (x, y).
Lemma continuous_snd {U V : UniformSpace} (x : U) (y : V) : continuous (snd (B:=V)) (x, y).
Lemma continuous_const {U V : UniformSpace} (c : V) (x : U) : continuous (fun _ => c) x.
Lemma continuous_id {U : UniformSpace} (x : U) : continuous (fun y => y) x.
Lemma continuous_opp (f : U -> V) (x : U) : continuous f x -> continuous (fun x : U => opp (f x)) 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.
Lemma continuous_minus (f g : U -> V) (x : U) : continuous f x -> continuous g x -> continuous (fun x : U => minus (f x) (g x)) x.
Lemma continuous_scal (k : U -> K) (f : U -> V) (x : U) : continuous k x -> continuous f x -> continuous (fun y => scal (k y) (f y)) x.
Lemma continuous_scal_r (k : K) (f : U -> V) (x : U) : continuous f x -> continuous (fun y => scal k (f y)) x.
Lemma continuous_scal_l (f : U -> K) (k : V) (x : U) : continuous f x -> continuous (fun y => scal (f y) k) x.
Lemma continuous_mult {U : UniformSpace} {K : AbsRing} (f g : U -> K) (x : U) : continuous f x -> continuous g x -> continuous (fun y => mult (f y) (g y)) x.
Lemma unifcont_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 (f x) eps (f y)}.
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)}.