(** Continuity and limit *)

(* Specifications: *)
Lemma is_lim_continuity (f : R -> R) (x : R) :
continuity_pt f x -> is_lim f x (f x).
Lemma ex_lim_continuity (f : R -> R) (x : R) :
continuity_pt f x -> ex_finite_lim f x.
Lemma Lim_continuity (f : R -> R) (x : R) :
continuity_pt f x -> Lim f x = f x.
Lemma C0_extension_right {T : UniformSpace} (f : R -> T) lb (a b : R) :
a < b ->
(forall c : R, a < c < b -> filterlim f (locally c) (locally (f c))) ->
(filterlim f (at_left b) (locally lb)) ->
{g : R -> T | (forall c, a < c -> filterlim g (locally c) (locally (g c)))
/\ (forall c : R, c < b -> g c = f c) /\ g b = lb}.
Lemma filterlim_Ropp_left (x : R) :
filterlim Ropp (at_left x) (at_right (- x)).
Lemma filterlim_Ropp_right (x : R) :
filterlim Ropp (at_right x) (at_left (- x)).
Lemma C0_extension_left {T : UniformSpace} (f : R -> T) la (a b : R) :
a < b ->
(forall c : R, a < c < b -> filterlim f (locally c) (locally (f c))) ->
(filterlim f (at_right a) (locally la)) ->
{g : R -> T | (forall c, c < b -> filterlim g (locally c) (locally (g c)))
/\ (forall c : R, a < c -> g c = f c) /\ g a = la}.
Lemma C0_extension_lt {T : UniformSpace} (f : R -> T) la lb (a b : R) :
a < b ->
(forall c : R, a < c < b -> filterlim f (locally c) (locally (f c))) ->
(filterlim f (at_right a) (locally la)) ->
(filterlim f (at_left b) (locally lb)) ->
{g : R -> T | (forall c, filterlim g (locally c) (locally (g c)))
/\ (forall c : R, a < c < b -> g c = f c) /\ g a = la /\ g b = lb}.
Lemma C0_extension_le {T : UniformSpace} (f : R -> T) (a b : R) :
(forall c : R, a <= c <= b -> filterlim f (locally c) (locally (f c))) ->
{g : R -> T | (forall c, filterlim g (locally c) (locally (g c)))
/\ (forall c : R, a <= c <= b -> g c = f c)}.
Lemma bounded_continuity {K : AbsRing} {V : NormedModule K}
(f : R -> V) a b :
(forall x, a <= x <= b -> filterlim f (locally x) (locally (f x)))
-> {M : R | forall x, a <= x <= b -> norm (f x) < M}.