Definition differentiable_pt (f : R -> R -> R) (x y : R) :=
exists lx, exists ly, differentiable_pt_lim f x y lx ly.

(* Specifications: *)
Lemma differentiable_continuity_pt : forall f x y,
differentiable_pt f x y -> continuity_2d_pt f x y.
Lemma differentiable_pt_lim_proj1_0 (f : R -> R) (x y l : R) :
derivable_pt_lim f x l -> differentiable_pt_lim (fun u v => f u) x y l 0.
Lemma differentiable_pt_lim_proj1_1 (f : R -> R) (x y l : R) :
differentiable_pt_lim (fun u v => f u) x y l 0 -> derivable_pt_lim f x l.
Lemma differentiable_pt_lim_unique (f : R -> R -> R) (x y : R) (lx ly : R) :
differentiable_pt_lim f x y lx ly
-> Derive (fun x => f x y) x = lx /\ Derive (fun y => f x y) y = ly.