/-- If a function `f` is differentiable in a convex open set and continuous on its closure, and its
derivative converges to a limit `f'` at a point on the boundary, then `f` is differentiable there
with derivative `f'`. -/
theorem has_fderiv_at_boundary_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
    (f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
    (f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
    (h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
    HasFDerivWithinAt f f' (closure s) x := by
  classical
    -- one can assume without loss of generality that `x` belongs to the closure of `s`, as the
    -- statement is empty otherwise
    by_cases hx : x ∉ closure s
    · rw [← closure_closure] at hx; exact hasFDerivWithinAt_of_nmem_closure hx
    push_neg at hx
    rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, Asymptotics.isLittleO_iff]
    /- One needs to show that `‖f y - f x - f' (y - x)‖ ≤ ε ‖y - x‖` for `y` close to `x` in
      `closure s`, where `ε` is an arbitrary positive constant. By continuity of the functions, it
      suffices to prove this for nearby points inside `s`. In a neighborhood of `x`, the derivative
      of `f` is arbitrarily close to `f'` by assumption. The mean value inequality completes the
      proof. -/
    intro ε ε_pos
    obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε := by
      simpa [dist_zero_right] using tendsto_nhdsWithin_nhds.1 h ε ε_pos
    set B := ball x δ
    suffices : ∀ y ∈ B ∩ closure s, ‖f y - f x - (f' y - f' x)‖ ≤ ε * ‖y - x‖
    exact mem_nhdsWithin_iff.2 ⟨δ, δ_pos, fun y hy => by simpa using this y hy⟩
    suffices
      ∀ p : E × E,
        p ∈ closure ((B ∩ s) ×ˢ (B ∩ s)) → ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖ by
      rw [closure_prod_eq] at this
      intro y y_in
      apply this ⟨x, y⟩
      have : B ∩ closure s ⊆ closure (B ∩ s) := isOpen_ball.inter_closure
      exact ⟨this ⟨mem_ball_self δ_pos, hx⟩, this y_in⟩
    have key : ∀ p : E × E, p ∈ (B ∩ s) ×ˢ (B ∩ s) →
          ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖ := by
      rintro ⟨u, v⟩ ⟨u_in, v_in⟩
      have conv : Convex ℝ (B ∩ s) := (convex_ball _ _).inter s_conv
      have diff : DifferentiableOn ℝ f (B ∩ s) := f_diff.mono (inter_subset_right _ _)
      have bound : ∀ z ∈ B ∩ s, ‖fderivWithin ℝ f (B ∩ s) z - f'‖ ≤ ε := by
        intro z z_in
        have h := hδ z
        have : fderivWithin ℝ f (B ∩ s) z = fderiv ℝ f z := by
          have op : IsOpen (B ∩ s) := isOpen_ball.inter s_open
          rw [DifferentiableAt.fderivWithin _ (op.uniqueDiffOn z z_in)]
          exact (diff z z_in).differentiableAt (IsOpen.mem_nhds op z_in)
        rw [← this] at h
        exact (le_of_lt (h z_in.2 z_in.1))
      simpa using conv.norm_image_sub_le_of_norm_fderivWithin_le' diff bound u_in v_in
    rintro ⟨u, v⟩ uv_in
    refine' ContinuousWithinAt.closure_le uv_in _ _ key
    have f_cont' : ∀ y ∈ closure s, ContinuousWithinAt (f -  ⇑f') s y := by
      intro y y_in
      exact Tendsto.sub (f_cont y y_in) f'.cont.continuousWithinAt
    

Proof state:
case this.mk.refine'_1
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y
h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure s, ContinuousWithinAt (f - ⇑f') s y
⊢ ContinuousWithinAt (fun y => ‖f y.2 - f y.1 - (f' y.2 - f' y.1)‖) ((B ∩ s) ×ˢ (B ∩ s)) (u, v)
case this.mk.refine'_2
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen s
f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y
h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure ((B ∩ s) ×ˢ (B ∩ s))
⊢ ContinuousWithinAt (fun y => ε * ‖y.2 - y.1‖) ((B ∩ s) ×ˢ (B ∩ s)) (u, v)

Next tactic:
all_goals
      -- common start for both continuity proofs
      have : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s := by mono <;> exact inter_subset_right _ _
      obtain ⟨u_in, v_in⟩ : u ∈ closure s ∧ v ∈ closure s := by
        simpa [closure_prod_eq] using closure_mono this uv_in
      apply ContinuousWithinAt.mono _ this
      simp only [ContinuousWithinAt]