End Continuity.

(* Specifications: *)
Lemma ex_RInt_locally {V : CompleteNormedModule R_AbsRing} (f : R -> V) (a b : R) :
ex_RInt f a b
-> (exists eps : posreal, ex_RInt f (a - eps) (a + eps))
-> (exists eps : posreal, ex_RInt f (b - eps) (b + eps))
-> locally (a,b) (fun z : R * R => ex_RInt f (fst z) (snd z)).
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.