-- Mathlib/Topology/Order/LeftRightLim.lean
theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x

Proof state:
case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousWithinAt f (Ioi x) x

Tactic:
rw [h] at h'

Result:
case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : rightLim f x = f x
⊢ ContinuousWithinAt f (Ioi x) x

Proof state:
case refine_2.refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousWithinAt f (Iio x) x

Tactic:
exact hf.continuousWithinAt_Iio_iff_leftLim_eq.2 h'

Result:
no goals

Proof state:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ ContinuousAt f x

Tactic:
have h' : leftLim f x = f x := by
  apply le_antisymm (leftLim_le hf (le_refl _))
  rw [h]
  exact le_rightLim hf (le_refl _)

Result:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousAt f x

Proof state:
case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : rightLim f x = f x
⊢ ContinuousWithinAt f (Ioi x) x

Tactic:
exact hf.continuousWithinAt_Ioi_iff_rightLim_eq.2 h'

Result:
no goals

Proof state:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
⊢ leftLim f x = rightLim f x

Tactic:
have A : leftLim f x = f x :=
  hf.continuousWithinAt_Iio_iff_leftLim_eq.1 h.continuousWithinAt

Result:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
A : leftLim f x = f x
⊢ leftLim f x = rightLim f x

Proof state:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
A : leftLim f x = f x
⊢ leftLim f x = rightLim f x

Tactic:
have B : rightLim f x = f x :=
  hf.continuousWithinAt_Ioi_iff_rightLim_eq.1 h.continuousWithinAt

Result:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
A : leftLim f x = f x
B : rightLim f x = f x
⊢ leftLim f x = rightLim f x

Proof state:
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ f x ≤ rightLim f x

Tactic:
exact le_rightLim hf (le_refl _)

Result:
no goals

Proof state:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
A : leftLim f x = f x
B : rightLim f x = f x
⊢ leftLim f x = rightLim f x

Tactic:
exact A.trans B.symm

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ ContinuousAt f x ↔ leftLim f x = rightLim f x

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Result:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : ContinuousAt f x
⊢ leftLim f x = rightLim f x

case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ ContinuousAt f x

Proof state:
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ leftLim f x = f x

Tactic:
apply le_antisymm (leftLim_le hf (le_refl _))

Result:
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ f x ≤ leftLim f x

Proof state:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousAt f x

Tactic:
refine continuousAt_iff_continuous_left'_right'.2 ⟨?_, ?_⟩

Result:
case refine_2.refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousWithinAt f (Iio x) x

case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
h' : leftLim f x = f x
⊢ ContinuousWithinAt f (Ioi x) x