English
For a convex f on S, the function x ↦ derivWithin f(Iio x) x is monotone on interior S.
Русский
Для выпуклой f на S функция x ↦ derivWithin f(Iio x) x монотонна на interior S.
LaTeX
$$$$ \ MonotoneOn\ (\lambda x.\ derivWithin\ f\ (Iio\ x)\ x)\ (\operatorname{interior} S) $$$$
Lean4
theorem monotoneOn_leftDeriv (hfc : ConvexOn ℝ S f) : MonotoneOn (fun x ↦ derivWithin f (Iio x) x) (interior S) :=
by
intro x hxs y hys hxy
rcases eq_or_lt_of_le hxy with rfl | hxy; · rfl
simp_rw [hfc.leftDeriv_eq_sSup_slope_of_mem_interior hxs, hfc.leftDeriv_eq_sSup_slope_of_mem_interior hys]
refine
le_csSup_of_le (b := slope f x y) (bddAbove_slope_gt_of_mem_interior hfc hys)
⟨x, by simp only [slope_comm, mem_setOf_eq, hxy, and_true]; exact interior_subset hxs⟩ (csSup_le ?_ ?_)
· have hxs' := hxs
rw [mem_interior_iff_mem_nhds, mem_nhds_iff_exists_Ioo_subset] at hxs'
obtain ⟨a, b, hxab, habs⟩ := hxs'
rw [image_nonempty]
obtain ⟨z, hxz, hzb⟩ := exists_between hxab.1
exact ⟨z, habs ⟨hxz, hzb.trans hxab.2⟩, hzb⟩
· rintro _ ⟨z, ⟨hzs, hyz : z < x⟩, rfl⟩
exact slope_mono hfc (interior_subset hxs) ⟨hzs, hyz.ne⟩ ⟨interior_subset hys, hxy.ne'⟩ (hyz.trans hxy).le
