-- Mathlib/Analysis/Calculus/MeanValue.lean
theorem not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi (f : ℝ → ℝ) {a : ℝ}
    (hf : Tendsto (deriv f) (𝓝[>] a) atTop) : ¬ DifferentiableWithinAt ℝ f (Ioi a) a

Proof state:
case pos
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
hcont_Ioc : ∀ z ∈ Ioc a b, ContinuousWithinAt f (Icc a b) z
z : ℝ
hz : z ∈ Icc a b
hz' : z = a
⊢ ContinuousWithinAt f (Icc a b) z

Tactic:
rw [hz']

Result:
case pos
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
hcont_Ioc : ∀ z ∈ Ioc a b, ContinuousWithinAt f (Icc a b) z
z : ℝ
hz : z ∈ Icc a b
hz' : z = a
⊢ ContinuousWithinAt f (Icc a b) a

Proof state:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
⊢ ¬DifferentiableWithinAt ℝ f (Ioi a) a

Tactic:
by_cases hcont_at_a : ContinuousWithinAt f (Ici a) a

Result:
case pos
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
⊢ ¬DifferentiableWithinAt ℝ f (Ioi a) a

case neg
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ¬ContinuousWithinAt f (Ici a) a
⊢ ¬DifferentiableWithinAt ℝ f (Ioi a) a

Proof state:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
z : ℝ
hz : z ∈ Ioc a b
⊢ DifferentiableWithinAt ℝ f (Ioi a) z

Tactic:
have : derivWithin f (Ioi a) z ≠ 0 := ne_of_gt <| by
  simp_all only [mem_Ioo, and_imp, mem_Ioc, max_lt_iff]

Result:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
z : ℝ
hz : z ∈ Ioc a b
this : derivWithin f (Ioi a) z ≠ 0
⊢ DifferentiableWithinAt ℝ f (Ioi a) z

Proof state:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff :
  ∀ (i : ℝ × ℝ),
    i.1 < derivWithin f (Ioi a) a ∧ derivWithin f (Ioi a) a < i.2 → ∀ᶠ (x : ℝ) in 𝓝[>] a, slope f a x ∈ Ioo i.1 i.2
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1

Tactic:
specialize hdiff ⟨derivWithin f (Ioi a) a - 1, derivWithin f (Ioi a) a + 1⟩ <| by simp

Result:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hdiff :
  ∀ᶠ (x : ℝ) in 𝓝[>] a,
    slope f a x ∈
      Ioo (derivWithin f (Ioi a) a - 1, derivWithin f (Ioi a) a + 1).1
        (derivWithin f (Ioi a) a - 1, derivWithin f (Ioi a) a + 1).2
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1

Proof state:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
hcont_Ioc : ∀ z ∈ Ioc a b, ContinuousWithinAt f (Icc a b) z
z : ℝ
hz : z ∈ Icc a b
⊢ ContinuousWithinAt f (Icc a b) z

Tactic:
by_cases hz' : z = a

Result:
case pos
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
hcont_Ioc : ∀ z ∈ Ioc a b, ContinuousWithinAt f (Icc a b) z
z : ℝ
hz : z ∈ Icc a b
hz' : z = a
⊢ ContinuousWithinAt f (Icc a b) z

case neg
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
hdiff' : DifferentiableOn ℝ f (Ioc a b)
hcont_Ioc : ∀ z ∈ Ioc a b, ContinuousWithinAt f (Icc a b) z
z : ℝ
hz : z ∈ Icc a b
hz' : ¬z = a
⊢ ContinuousWithinAt f (Icc a b) z