File: Mathlib/Analysis/BoundedVariation.lean

full_name: eVariationOn
code: /-- The (extended real valued) variation of a function `f` on a set `s` inside a linear order is
the supremum of the sum of `edist (f (u (i+1))) (f (u i))` over all finite increasing
sequences `u` in `s`. -/
noncomputable def eVariationOn (f : α → E) (s : Set α) : ℝ≥0∞ 
goal: ℝ≥0∞

full_name: BoundedVariationOn
code: /-- A function has bounded variation on a set `s` if its total variation there is finite. -/
def BoundedVariationOn (f : α → E) (s : Set α) 
goal: Prop

full_name: LocallyBoundedVariationOn
code: /-- A function has locally bounded variation on a set `s` if, given any interval `[a, b]` with
endpoints in `s`, then the function has finite variation on `s ∩ [a, b]`. -/
def LocallyBoundedVariationOn (f : α → E) (s : Set α) 
goal: Prop

full_name: eVariationOn.nonempty_monotone_mem
code: theorem nonempty_monotone_mem {s : Set α} (hs : s.Nonempty) :
    Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } 
goal: Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }

full_name: eVariationOn.eq_of_edist_zero_on
code: theorem eq_of_edist_zero_on {f f' : α → E} {s : Set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) :
    eVariationOn f s = eVariationOn f' s 
goal: eVariationOn f s = eVariationOn f' s

full_name: eVariationOn.eq_of_eqOn
code: theorem eq_of_eqOn {f f' : α → E} {s : Set α} (h : EqOn f f' s) :
    eVariationOn f s = eVariationOn f' s 
goal: eVariationOn f s = eVariationOn f' s

full_name: eVariationOn.sum_le
code: theorem sum_le (f : α → E) {s : Set α} (n : ℕ) {u : ℕ → α} (hu : Monotone u) (us : ∀ i, u i ∈ s) :
    (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
goal: ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

full_name: eVariationOn.sum_le_of_monotoneOn_Icc
code: theorem sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α}
    (hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) :
    (∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
goal: ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

full_name: eVariationOn.sum_le_of_monotoneOn_Iic
code: theorem sum_le_of_monotoneOn_Iic (f : α → E) {s : Set α} {n : ℕ} {u : ℕ → α}
    (hu : MonotoneOn u (Iic n)) (us : ∀ i ≤ n, u i ∈ s) :
    (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
goal: ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

full_name: eVariationOn.mono
code: theorem mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s 
goal: eVariationOn f t ≤ eVariationOn f s

full_name: BoundedVariationOn.mono
code: theorem _root_.BoundedVariationOn.mono {f : α → E} {s : Set α} (h : BoundedVariationOn f s)
    {t : Set α} (ht : t ⊆ s) : BoundedVariationOn f t 
goal: BoundedVariationOn f t

full_name: BoundedVariationOn.locallyBoundedVariationOn
code: theorem _root_.BoundedVariationOn.locallyBoundedVariationOn {f : α → E} {s : Set α}
    (h : BoundedVariationOn f s) : LocallyBoundedVariationOn f s 
goal: LocallyBoundedVariationOn f s

full_name: eVariationOn.edist_le
code: theorem edist_le (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
    edist (f x) (f y) ≤ eVariationOn f s 
goal: edist (f x) (f y) ≤ eVariationOn f s

full_name: eVariationOn.eq_zero_iff
code: theorem eq_zero_iff (f : α → E) {s : Set α} :
    eVariationOn f s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0 
goal: eVariationOn f s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0

full_name: eVariationOn.constant_on
code: theorem constant_on {f : α → E} {s : Set α} (hf : (f '' s).Subsingleton) :
    eVariationOn f s = 0 
goal: eVariationOn f s = 0

full_name: eVariationOn.subsingleton
code: @[simp]
protected theorem subsingleton (f : α → E) {s : Set α} (hs : s.Subsingleton) :
    eVariationOn f s = 0 
goal: eVariationOn f s = 0

full_name: eVariationOn.lowerSemicontinuous_aux
code: theorem lowerSemicontinuous_aux {ι : Type*} {F : ι → α → E} {p : Filter ι} {f : α → E} {s : Set α}
    (Ffs : ∀ x ∈ s, Tendsto (fun i => F i x) p (𝓝 (f x))) {v : ℝ≥0∞} (hv : v < eVariationOn f s) :
    ∀ᶠ n : ι in p, v < eVariationOn (F n) s 
goal: ∀ᶠ (n : ι) in p, v < eVariationOn (F n) s

full_name: eVariationOn.lowerSemicontinuous
code: /-- The map `(eVariationOn · s)` is lower semicontinuous for pointwise convergence *on `s`*.
Pointwise convergence on `s` is encoded here as uniform convergence on the family consisting of the
singletons of elements of `s`.
-/
protected theorem lowerSemicontinuous (s : Set α) :
    LowerSemicontinuous fun f : α →ᵤ[s.image singleton] E => eVariationOn f s 
goal: LowerSemicontinuous fun f => eVariationOn f s

full_name: eVariationOn.lowerSemicontinuous_uniformOn
code: /-- The map `(eVariationOn · s)` is lower semicontinuous for uniform convergence on `s`.  -/
theorem lowerSemicontinuous_uniformOn (s : Set α) :
    LowerSemicontinuous fun f : α →ᵤ[{s}] E => eVariationOn f s 
goal: LowerSemicontinuous fun f => eVariationOn f s

full_name: BoundedVariationOn.dist_le
code: theorem _root_.BoundedVariationOn.dist_le {E : Type*} [PseudoMetricSpace E] {f : α → E}
    {s : Set α} (h : BoundedVariationOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
    dist (f x) (f y) ≤ (eVariationOn f s).toReal 
goal: dist (f x) (f y) ≤ (eVariationOn f s).toReal

full_name: BoundedVariationOn.sub_le
code: theorem _root_.BoundedVariationOn.sub_le {f : α → ℝ} {s : Set α} (h : BoundedVariationOn f s)
    {x y : α} (hx : x ∈ s) (hy : y ∈ s) : f x - f y ≤ (eVariationOn f s).toReal 
goal: f x - f y ≤ (eVariationOn f s).toReal

full_name: eVariationOn.add_point
code: /-- Consider a monotone function `u` parameterizing some points of a set `s`. Given `x ∈ s`, then
one can find another monotone function `v` parameterizing the same points as `u`, with `x` added.
In particular, the variation of a function along `u` is bounded by its variation along `v`. -/
theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
    (us : ∀ i, u i ∈ s) (n : ℕ) :
    ∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
      (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
        ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j)) 
goal: ∃ v m,
  Monotone v ∧
    (∀ (i : ℕ), v i ∈ s) ∧
      x ∈ v '' Set.Iio m ∧
        ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))

full_name: eVariationOn.add_le_union
code: /-- The variation of a function on the union of two sets `s` and `t`, with `s` to the left of `t`,
bounds the sum of the variations along `s` and `t`. -/
theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
    eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) 
goal: eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)

full_name: eVariationOn.union
code: /-- If a set `s` is to the left of a set `t`, and both contain the boundary point `x`, then
the variation of `f` along `s ∪ t` is the sum of the variations. -/
theorem union (f : α → E) {s t : Set α} {x : α} (hs : IsGreatest s x) (ht : IsLeast t x) :
    eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t 
goal: eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t

full_name: eVariationOn.Icc_add_Icc
code: theorem Icc_add_Icc (f : α → E) {s : Set α} {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) (hb : b ∈ s) :
    eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c) 
goal: eVariationOn f (s ∩ Set.Icc a b) + eVariationOn f (s ∩ Set.Icc b c) = eVariationOn f (s ∩ Set.Icc a c)

full_name: eVariationOn.comp_le_of_monotoneOn
code: theorem comp_le_of_monotoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
    (φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s 
goal: eVariationOn (f ∘ φ) t ≤ eVariationOn f s

full_name: eVariationOn.comp_le_of_antitoneOn
code: theorem comp_le_of_antitoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
    (φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s 
goal: eVariationOn (f ∘ φ) t ≤ eVariationOn f s

full_name: eVariationOn.comp_eq_of_monotoneOn
code: theorem comp_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) :
    eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) 
goal: eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

full_name: eVariationOn.comp_inter_Icc_eq_of_monotoneOn
code: theorem comp_inter_Icc_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
    {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
    eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y)) 
goal: eVariationOn (f ∘ φ) (t ∩ Set.Icc x y) = eVariationOn f (φ '' t ∩ Set.Icc (φ x) (φ y))

full_name: eVariationOn.comp_eq_of_antitoneOn
code: theorem comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
    eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) 
goal: eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)

full_name: eVariationOn.comp_ofDual
code: theorem comp_ofDual (f : α → E) (s : Set α) :
    eVariationOn (f ∘ ofDual) (ofDual ⁻¹' s) = eVariationOn f s 
goal: eVariationOn (f ∘ ⇑OrderDual.ofDual) (⇑OrderDual.ofDual ⁻¹' s) = eVariationOn f s

full_name: MonotoneOn.eVariationOn_le
code: theorem MonotoneOn.eVariationOn_le {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) {a b : α}
    (as : a ∈ s) (bs : b ∈ s) : eVariationOn f (s ∩ Icc a b) ≤ ENNReal.ofReal (f b - f a) 
goal: eVariationOn f (s ∩ Set.Icc a b) ≤ ENNReal.ofReal (f b - f a)

full_name: MonotoneOn.locallyBoundedVariationOn
code: theorem MonotoneOn.locallyBoundedVariationOn {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) :
    LocallyBoundedVariationOn f s 
goal: LocallyBoundedVariationOn f s

full_name: variationOnFromTo
code: /-- The **signed** variation of `f` on the interval `Icc a b` intersected with the set `s`,
squashed to a real (therefore only really meaningful if the variation is finite)
-/
noncomputable def variationOnFromTo (f : α → E) (s : Set α) (a b : α) : ℝ 
goal: ℝ

full_name: variationOnFromTo.self
code: protected theorem self (a : α) : variationOnFromTo f s a a = 0 
goal: variationOnFromTo f s a a = 0

full_name: variationOnFromTo.nonneg_of_le
code: protected theorem nonneg_of_le {a b : α} (h : a ≤ b) : 0 ≤ variationOnFromTo f s a b 
goal: 0 ≤ variationOnFromTo f s a b

full_name: variationOnFromTo.eq_neg_swap
code: protected theorem eq_neg_swap (a b : α) :
    variationOnFromTo f s a b = -variationOnFromTo f s b a 
goal: variationOnFromTo f s a b = -variationOnFromTo f s b a

full_name: variationOnFromTo.nonpos_of_ge
code: protected theorem nonpos_of_ge {a b : α} (h : b ≤ a) : variationOnFromTo f s a b ≤ 0 
goal: variationOnFromTo f s a b ≤ 0

full_name: variationOnFromTo.eq_of_le
code: protected theorem eq_of_le {a b : α} (h : a ≤ b) :
    variationOnFromTo f s a b = (eVariationOn f (s ∩ Icc a b)).toReal 
goal: variationOnFromTo f s a b = (eVariationOn f (s ∩ Set.Icc a b)).toReal

full_name: variationOnFromTo.eq_of_ge
code: protected theorem eq_of_ge {a b : α} (h : b ≤ a) :
    variationOnFromTo f s a b = -(eVariationOn f (s ∩ Icc b a)).toReal 
goal: variationOnFromTo f s a b = -(eVariationOn f (s ∩ Set.Icc b a)).toReal

full_name: variationOnFromTo.add
code: protected theorem add {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b c : α}
    (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
    variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c 
goal: variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c

full_name: variationOnFromTo.edist_zero_of_eq_zero
code: protected theorem edist_zero_of_eq_zero {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : variationOnFromTo f s a b = 0) :
    edist (f a) (f b) = 0 
goal: edist (f a) (f b) = 0

full_name: variationOnFromTo.eq_left_iff
code: protected theorem eq_left_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
    variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0 
goal: variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0

full_name: variationOnFromTo.eq_zero_iff_of_le
code: protected theorem eq_zero_iff_of_le {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ab : a ≤ b) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc a b) ⦃y⦄ (_hy : y ∈ s ∩ Icc a b), edist (f x) (f y) = 0 
goal: variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Set.Icc a b → ∀ ⦃y : α⦄, y ∈ s ∩ Set.Icc a b → edist (f x) (f y) = 0

full_name: variationOnFromTo.eq_zero_iff_of_ge
code: protected theorem eq_zero_iff_of_ge {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ba : b ≤ a) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc b a) ⦃y⦄ (_hy : y ∈ s ∩ Icc b a), edist (f x) (f y) = 0 
goal: variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Set.Icc b a → ∀ ⦃y : α⦄, y ∈ s ∩ Set.Icc b a → edist (f x) (f y) = 0

full_name: variationOnFromTo.eq_zero_iff
code: protected theorem eq_zero_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α}
    (ha : a ∈ s) (hb : b ∈ s) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ uIcc a b) ⦃y⦄ (_hy : y ∈ s ∩ uIcc a b), edist (f x) (f y) = 0 
goal: variationOnFromTo f s a b = 0 ↔
  ∀ ⦃x : α⦄, x ∈ s ∩ Set.uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ Set.uIcc a b → edist (f x) (f y) = 0

full_name: variationOnFromTo.monotoneOn
code: protected theorem monotoneOn (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
    MonotoneOn (variationOnFromTo f s a) s 
goal: MonotoneOn (variationOnFromTo f s a) s

full_name: variationOnFromTo.antitoneOn
code: protected theorem antitoneOn (hf : LocallyBoundedVariationOn f s) {b : α} (bs : b ∈ s) :
    AntitoneOn (fun a => variationOnFromTo f s a b) s 
goal: AntitoneOn (fun a => variationOnFromTo f s a b) s

full_name: variationOnFromTo.sub_self_monotoneOn
code: protected theorem sub_self_monotoneOn {f : α → ℝ} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a : α} (as : a ∈ s) : MonotoneOn (variationOnFromTo f s a - f) s 
goal: MonotoneOn (variationOnFromTo f s a - f) s

full_name: variationOnFromTo.comp_eq_of_monotoneOn
code: protected theorem comp_eq_of_monotoneOn {β : Type*} [LinearOrder β] (f : α → E) {t : Set β}
    (φ : β → α) (hφ : MonotoneOn φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
    variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y) 
goal: variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y)

full_name: LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn
code: /-- If a real valued function has bounded variation on a set, then it is a difference of monotone
functions there. -/
theorem LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn {f : α → ℝ} {s : Set α}
    (h : LocallyBoundedVariationOn f s) :
    ∃ p q : α → ℝ, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q 
goal: ∃ p q, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q

full_name: LipschitzOnWith.comp_eVariationOn_le
code: theorem LipschitzOnWith.comp_eVariationOn_le {f : E → F} {C : ℝ≥0} {t : Set E}
    (h : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t) :
    eVariationOn (f ∘ g) s ≤ C * eVariationOn g s 
goal: eVariationOn (f ∘ g) s ≤ ↑C * eVariationOn g s

full_name: LipschitzOnWith.comp_boundedVariationOn
code: theorem LipschitzOnWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
    (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
    (h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s 
goal: BoundedVariationOn (f ∘ g) s

full_name: LipschitzOnWith.comp_locallyBoundedVariationOn
code: theorem LipschitzOnWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
    (hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
    (h : LocallyBoundedVariationOn g s) : LocallyBoundedVariationOn (f ∘ g) s 
goal: LocallyBoundedVariationOn (f ∘ g) s

full_name: LipschitzWith.comp_boundedVariationOn
code: theorem LipschitzWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} (hf : LipschitzWith C f)
    {g : α → E} {s : Set α} (h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s 
goal: BoundedVariationOn (f ∘ g) s

full_name: LipschitzWith.comp_locallyBoundedVariationOn
code: theorem LipschitzWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0}
    (hf : LipschitzWith C f) {g : α → E} {s : Set α} (h : LocallyBoundedVariationOn g s) :
    LocallyBoundedVariationOn (f ∘ g) s 
goal: LocallyBoundedVariationOn (f ∘ g) s

full_name: LipschitzOnWith.locallyBoundedVariationOn
code: theorem LipschitzOnWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} {s : Set ℝ}
    (hf : LipschitzOnWith C f s) : LocallyBoundedVariationOn f s 
goal: LocallyBoundedVariationOn f s

full_name: LipschitzWith.locallyBoundedVariationOn
code: theorem LipschitzWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} (hf : LipschitzWith C f)
    (s : Set ℝ) : LocallyBoundedVariationOn f s 
goal: LocallyBoundedVariationOn f s

full_name: LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_real
code: /-- A bounded variation function into `ℝ` is differentiable almost everywhere. Superseded by
`ae_differentiableWithinAt_of_mem`. -/
theorem ae_differentiableWithinAt_of_mem_real {f : ℝ → ℝ} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

full_name: LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_pi
code: /-- A bounded variation function into a finite dimensional product vector space is differentiable
almost everywhere. Superseded by `ae_differentiableWithinAt_of_mem`. -/
theorem ae_differentiableWithinAt_of_mem_pi {ι : Type*} [Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

full_name: LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem
code: /-- A real function into a finite dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. -/
theorem ae_differentiableWithinAt_of_mem {f : ℝ → V} {s : Set ℝ}
    (h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

full_name: LocallyBoundedVariationOn.ae_differentiableWithinAt
code: /-- A real function into a finite dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. -/
theorem ae_differentiableWithinAt {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
    (hs : MeasurableSet s) : ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x

full_name: LocallyBoundedVariationOn.ae_differentiableAt
code: /-- A real function into a finite dimensional real vector space with bounded variation
is differentiable almost everywhere. -/
theorem ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x

full_name: LipschitzOnWith.ae_differentiableWithinAt_of_mem_real
code: /-- A real function into a finite dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzOnWith.ae_differentiableWithinAt_of_mem`.
-/
theorem LipschitzOnWith.ae_differentiableWithinAt_of_mem_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
    (h : LipschitzOnWith C f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

full_name: LipschitzOnWith.ae_differentiableWithinAt_real
code: /-- A real function into a finite dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzOnWith.ae_differentiableWithinAt`. -/
theorem LipschitzOnWith.ae_differentiableWithinAt_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
    (h : LipschitzOnWith C f s) (hs : MeasurableSet s) :
    ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x

full_name: LipschitzWith.ae_differentiableAt_real
code: /-- A real Lipschitz function into a finite dimensional real vector space is differentiable
almost everywhere. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzWith.ae_differentiableAt`. -/
theorem LipschitzWith.ae_differentiableAt_real {C : ℝ≥0} {f : ℝ → V} (h : LipschitzWith C f) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x