File: Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean

full_name: not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
code: /-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k 
goal: ¬MeasureTheory.IntegrableOn g k MeasureTheory.volume

full_name: not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
code: theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k 
goal: ¬MeasureTheory.IntegrableOn g k MeasureTheory.volume

full_name: not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter
code: /-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any interval `a..b` such that
`[a, b] ∈ l`. -/
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter {f : ℝ → E} {g : ℝ → F}
    {a b : ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l] (hl : [[a, b]] ∈ l)
    (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntervalIntegrable g volume a b 
goal: ¬IntervalIntegrable g MeasureTheory.volume a b

full_name: not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
code: /-- If `a ≠ b`, `c ∈ [a, b]`, `f` is differentiable in the neighborhood of `c` within
`[a, b] \ {c}`, `‖f x‖ → ∞` as `x → c` within `[a, b] \ {c}`, and `f' = O(g)` along
`𝓝[[a, b] \ {c}] c`, where `f'` is the derivative of `f`, then `g` is not interval integrable on
`a..b`. -/
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
    {f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (hne : a ≠ b) (hc : c ∈ [[a, b]])
    (h_deriv : ∀ᶠ x in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x)
    (h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop)
    (hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g) : ¬IntervalIntegrable g volume a b 
goal: ¬IntervalIntegrable g MeasureTheory.volume a b

full_name: not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured
code: /-- If `f` is differentiable in a punctured neighborhood of `c`, `‖f x‖ → ∞` as `x → c` (more
formally, along the filter `𝓝[≠] c`), and `f' = O(g)` along `𝓝[≠] c`, where `f'` is the derivative
of `f`, then `g` is not interval integrable on any nontrivial interval `a..b` such that
`c ∈ [a, b]`. -/
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured {f : ℝ → E}
    {g : ℝ → F} {a b c : ℝ} (h_deriv : ∀ᶠ x in 𝓝[≠] c, DifferentiableAt ℝ f x)
    (h_infty : Tendsto (fun x => ‖f x‖) (𝓝[≠] c) atTop) (hg : deriv f =O[𝓝[≠] c] g) (hne : a ≠ b)
    (hc : c ∈ [[a, b]]) : ¬IntervalIntegrable g volume a b 
goal: ¬IntervalIntegrable g MeasureTheory.volume a b

full_name: not_intervalIntegrable_of_sub_inv_isBigO_punctured
code: /-- If `f` grows in the punctured neighborhood of `c : ℝ` at least as fast as `1 / (x - c)`,
then it is not interval integrable on any nontrivial interval `a..b`, `c ∈ [a, b]`. -/
theorem not_intervalIntegrable_of_sub_inv_isBigO_punctured {f : ℝ → F} {a b c : ℝ}
    (hf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f) (hne : a ≠ b) (hc : c ∈ [[a, b]]) :
    ¬IntervalIntegrable f volume a b 
goal: ¬IntervalIntegrable f MeasureTheory.volume a b

full_name: intervalIntegrable_sub_inv_iff
code: /-- The function `fun x => (x - c)⁻¹` is integrable on `a..b` if and only if
`a = b` or `c ∉ [a, b]`. -/
@[simp]
theorem intervalIntegrable_sub_inv_iff {a b c : ℝ} :
    IntervalIntegrable (fun x => (x - c)⁻¹) volume a b ↔ a = b ∨ c ∉ [[a, b]] 
goal: IntervalIntegrable (fun x => (x - c)⁻¹) MeasureTheory.volume a b ↔ a = b ∨ c ∉ [[a, b]]

full_name: intervalIntegrable_inv_iff
code: /-- The function `fun x => x⁻¹` is integrable on `a..b` if and only if
`a = b` or `0 ∉ [a, b]`. -/
@[simp]
theorem intervalIntegrable_inv_iff {a b : ℝ} :
    IntervalIntegrable (fun x => x⁻¹) volume a b ↔ a = b ∨ (0 : ℝ) ∉ [[a, b]] 
goal: IntervalIntegrable (fun x => x⁻¹) MeasureTheory.volume a b ↔ a = b ∨ 0 ∉ [[a, b]]

full_name: not_IntegrableOn_Ici_inv
code: /-- The function `fun x ↦ x⁻¹` is not integrable on any interval `[a, +∞)`. -/
theorem not_IntegrableOn_Ici_inv {a : ℝ} :
    ¬ IntegrableOn (fun x => x⁻¹) (Ici a) 
goal: ¬MeasureTheory.IntegrableOn (fun x => x⁻¹) (Set.Ici a) MeasureTheory.volume

full_name: not_IntegrableOn_Ioi_inv
code: /-- The function `fun x ↦ x⁻¹` is not integrable on any interval `(a, +∞)`. -/
theorem not_IntegrableOn_Ioi_inv {a : ℝ} :
    ¬ IntegrableOn (·⁻¹) (Ioi a) 
goal: ¬MeasureTheory.IntegrableOn (fun x => x⁻¹) (Set.Ioi a) MeasureTheory.volume