theorem continuousWithinAt_primitive (hb₀ : μ {b₀} = 0)
    (h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)) :
    ContinuousWithinAt (fun b => ∫ x in a..b, f x ∂μ) (Icc b₁ b₂) b₀ := by
  by_cases h₀ : b₀ ∈ Icc b₁ b₂
  · have h₁₂ : b₁ ≤ b₂ := h₀.1.trans h₀.2
    have min₁₂ : min b₁ b₂ = b₁ := min_eq_left h₁₂
    have h_int' : ∀ {x}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x := by
      rintro x ⟨h₁, h₂⟩
      apply h_int.mono_set
      apply uIcc_subset_uIcc
      · exact ⟨min_le_of_left_le (min_le_right a b₁),
          h₁.trans (h₂.trans <| le_max_of_le_right <| le_max_right _ _)⟩
      · exact ⟨min_le_of_left_le <| (min_le_right _ _).trans h₁,
          le_max_of_le_right <| h₂.trans <| le_max_right _ _⟩
    have : ∀ b ∈ Icc b₁ b₂,
        ∫ x in a..b, f x ∂μ = (∫ x in a..b₁, f x ∂μ) + ∫ x in b₁..b, f x ∂μ := by
      rintro b ⟨h₁, h₂⟩
      rw [← integral_add_adjacent_intervals _ (h_int' ⟨h₁, h₂⟩)]
      apply h_int.mono_set
      apply uIcc_subset_uIcc
      · exact ⟨min_le_of_left_le (min_le_left a b₁), le_max_of_le_right (le_max_left _ _)⟩
      · exact ⟨min_le_of_left_le (min_le_right _ _),
          le_max_of_le_right (h₁.trans <| h₂.trans (le_max_right a b₂))⟩
    apply ContinuousWithinAt.congr _ this (this _ h₀); clear this
    refine' continuousWithinAt_const.add _
    have :
      (fun b => ∫ x in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b =>
        ∫ x in b₁..b₂, indicator {x | x ≤ b} f x ∂μ := by
      apply eventuallyEq_of_mem self_mem_nhdsWithin
      exact fun b b_in => (integral_indicator b_in).symm
    apply ContinuousWithinAt.congr_of_eventuallyEq _ this (integral_indicator h₀).symm
    have : IntervalIntegrable (fun x => ‖f x‖) μ b₁ b₂ :=
      IntervalIntegrable.norm (h_int' <| right_mem_Icc.mpr h₁₂)
    refine' continuousWithinAt_of_dominated_interval _ _ this _ <;> clear this
    · apply Eventually.mono self_mem_nhdsWithin
      intro x hx
      erw [aestronglyMeasurable_indicator_iff, Measure.restrict_restrict, Iic_inter_Ioc_of_le]
      · rw [min₁₂]
        exact (h_int' hx).1.aestronglyMeasurable
      · exact le_max_of_le_right hx.2
      exacts [measurableSet_Iic, measurableSet_Iic]
    · refine' eventually_of_forall fun x => eventually_of_forall fun t => _
      dsimp [indicator]
      split_ifs <;> simp
    · have : ∀ᵐ t ∂μ, t < b₀ ∨ b₀ < t := by
        apply Eventually.mono (compl_mem_ae_iff.mpr hb₀)
        intro x hx
        exact Ne.lt_or_lt hx
      apply this.mono
      rintro x₀ (hx₀ | hx₀) -
      · have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀ := by
          apply mem_nhdsWithin_of_mem_nhds
          apply Eventually.mono (Ioi_mem_nhds hx₀)
          intro x hx
          simp [hx.le]
        apply continuousWithinAt_const.congr_of_eventuallyEq this
        simp [hx₀.le]
      · have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = 0 := by
          apply mem_nhdsWithin_of_mem_nhds
          

Proof state:
case h
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a b b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f g : ℝ → E
hb₀ : ↑↑μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b => ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b => ∫ (x : ℝ) in b₁..b₂, indicator {x | x ≤ b} f x ∂μ
this : ∀ᵐ (t : ℝ) ∂μ, t < b₀ ∨ b₀ < t
x₀ : ℝ
hx₀ : b₀ < x₀
⊢ {x | (fun x => indicator {t | t ≤ x} f x₀ = 0) x} ∈ 𝓝 b₀

Next tactic:
apply Eventually.mono (Iio_mem_nhds hx₀)