(** * Differentiability in 1 dimentional space *)
Section Derive.
Context {K : AbsRing} {V : NormedModule K}.
Definition is_derive (f : K -> V) (x : K) (l : V) :=
filterdiff f (locally x) (fun y : K => scal y l).
Definition ex_derive (f : K -> V) (x : K) :=
exists l : V, is_derive f x l.

(* Specifications: *)
Lemma ex_derive_filterdiff :
forall (f : K -> V) (x : K),
ex_derive f x <-> ex_filterdiff f (locally x).
Lemma ex_derive_continuous (f : K -> V) (x : K) :
ex_derive f x -> continuous f x.