Section Continuity_op.
Context {U : UniformSpace} {K : AbsRing} {V : NormedModule K}.

(* Specifications: *)
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.