ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousOn.restrict_mapsTo
theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s)
    (ht : MapsTo f s t) : Continuous (ht.restrict f s t) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s) (ht : Set.MapsTo f s t)
Goal: Continuous (Set.MapsTo.restrict f s t ht)
-- continuousOn_iff'
theorem continuousOn_iff' {f : α → β} {s : Set α} :
    ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α}
Goal: ContinuousOn f s ↔ ∀ (t : Set β), IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s
-- ContinuousOn.mono_dom
theorem ContinuousOn.mono_dom {α β : Type*} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
    (h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₃ f s) :
    @ContinuousOn α β t₂ t₃ f s 
Arguments: {α : Type u_7} {β : Type u_8} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : ContinuousOn f s)
Goal: ContinuousOn f s
-- ContinuousOn.mono_rng
theorem ContinuousOn.mono_rng {α β : Type*} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
    (h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₂ f s) :
    @ContinuousOn α β t₁ t₃ f s 
Arguments: {α : Type u_7} {β : Type u_8} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : ContinuousOn f s)
Goal: ContinuousOn f s
-- continuousOn_iff_isClosed
theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
    ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α}
Goal: ContinuousOn f s ↔ ∀ (t : Set β), IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s
-- ContinuousOn.prod_map
theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β}
    (hf : ContinuousOn f s) (hg : ContinuousOn g t) : ContinuousOn (Prod.map f g) (s ×ˢ t) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} (hf : ContinuousOn f s) (hg : ContinuousOn g t)
Goal: ContinuousOn (Prod.map f g) (s ×ˢ t)
-- continuous_of_cover_nhds
theorem continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → Set α}
    (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) :
    Continuous f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Sort u_7} {f : α → β} {s : ι → Set α} (hs : ∀ (x : α), ∃ i, s i ∈ 𝓝 x) (hf : ∀ (i : ι), ContinuousOn f (s i))
Goal: Continuous f
-- continuousOn_empty
theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β)
Goal: ContinuousOn f ∅
-- continuousOn_singleton
@[simp]
theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (a : α)
Goal: ContinuousOn f {a}
-- Set.Subsingleton.continuousOn
theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α → β) :
    ContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} (hs : s.Subsingleton) (f : α → β)
Goal: ContinuousOn f s
-- nhdsWithin_le_comap
theorem nhdsWithin_le_comap {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x) :
    𝓝[s] x ≤ comap f (𝓝[f '' s] f x) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x)
Goal: 𝓝[s] x ≤ Filter.comap f (𝓝[f '' s] f x)
-- ContinuousWithinAt.mono
theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x)
    (hs : s ⊆ t) : ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x) (hs : s ⊆ t)
Goal: ContinuousWithinAt f s x
-- ContinuousWithinAt.mono_of_mem
theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
    (h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x) : ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x)
Goal: ContinuousWithinAt f s x
-- continuousWithinAt_congr_nhds
theorem continuousWithinAt_congr_nhds {f : α → β} {s t : Set α} {x : α} (h : 𝓝[s] x = 𝓝[t] x) :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (h : 𝓝[s] x = 𝓝[t] x)
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x
-- continuousWithinAt_inter'
theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) :
    ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x)
Goal: ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x
-- continuousWithinAt_inter
theorem continuousWithinAt_inter {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x) :
    ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x)
Goal: ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x
-- continuousWithinAt_union
theorem continuousWithinAt_union {f : α → β} {s t : Set α} {x : α} :
    ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α}
Goal: ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x
-- ContinuousWithinAt.union
theorem ContinuousWithinAt.union {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x)
    (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s ∪ t) x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x) (ht : ContinuousWithinAt f t x)
Goal: ContinuousWithinAt f (s ∪ t) x
-- ContinuousWithinAt.mem_closure_image
theorem ContinuousWithinAt.mem_closure_image {f : α → β} {s : Set α} {x : α}
    (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s)
Goal: f x ∈ closure (f '' s)
-- ContinuousWithinAt.mem_closure
theorem ContinuousWithinAt.mem_closure {f : α → β} {s : Set α} {x : α} {A : Set β}
    (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : MapsTo f s A) : f x ∈ closure A 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} {A : Set β} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : Set.MapsTo f s A)
Goal: f x ∈ closure A
-- Set.MapsTo.closure_of_continuousWithinAt
theorem Set.MapsTo.closure_of_continuousWithinAt {f : α → β} {s : Set α} {t : Set β}
    (h : MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x) :
    MapsTo f (closure s) (closure t) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {t : Set β} (h : Set.MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x)
Goal: Set.MapsTo f (closure s) (closure t)
-- Set.MapsTo.closure_of_continuousOn
theorem Set.MapsTo.closure_of_continuousOn {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t)
    (hc : ContinuousOn f (closure s)) : MapsTo f (closure s) (closure t) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {t : Set β} (h : Set.MapsTo f s t) (hc : ContinuousOn f (closure s))
Goal: Set.MapsTo f (closure s) (closure t)
-- ContinuousWithinAt.image_closure
theorem ContinuousWithinAt.image_closure {f : α → β} {s : Set α}
    (hf : ∀ x ∈ closure s, ContinuousWithinAt f s x) : f '' closure s ⊆ closure (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : ∀ x ∈ closure s, ContinuousWithinAt f s x)
Goal: f '' closure s ⊆ closure (f '' s)
-- ContinuousOn.image_closure
theorem ContinuousOn.image_closure {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s)) :
    f '' closure s ⊆ closure (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s))
Goal: f '' closure s ⊆ closure (f '' s)
-- continuousWithinAt_singleton
@[simp]
theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithinAt f {x} x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α}
Goal: ContinuousWithinAt f {x} x
-- continuousWithinAt_insert_self
@[simp]
theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
    ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {s : Set α}
Goal: ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x
-- ContinuousWithinAt.diff_iff
theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α}
    (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} {x : α} (ht : ContinuousWithinAt f t x)
Goal: ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x
-- continuousWithinAt_diff_self
@[simp]
theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} :
    ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α}
Goal: ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x
-- continuousWithinAt_compl_self
@[simp]
theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
    ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {a : α}
Goal: ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a
-- continuousWithinAt_update_same
@[simp]
theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} :
    ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β}
Goal: ContinuousWithinAt (Function.update f x y) s x ↔ Filter.Tendsto f (𝓝[s \ {x}] x) (𝓝 y)
-- continuousAt_update_same
@[simp]
theorem continuousAt_update_same [DecidableEq α] {f : α → β} {x : α} {y : β} :
    ContinuousAt (Function.update f x y) x ↔ Tendsto f (𝓝[≠] x) (𝓝 y) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [DecidableEq α] {f : α → β} {x : α} {y : β}
Goal: ContinuousAt (Function.update f x y) x ↔ Filter.Tendsto f (𝓝[≠] x) (𝓝 y)
-- IsOpenMap.continuousOn_image_of_leftInvOn
theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
    (h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : LeftInvOn finv f s) :
    ContinuousOn finv (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : Set.LeftInvOn finv f s)
Goal: ContinuousOn finv (f '' s)
-- IsOpenMap.continuousOn_range_of_leftInverse
theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
    (hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {finv : β → α} (hleft : Function.LeftInverse finv f)
Goal: ContinuousOn finv (Set.range f)
-- ContinuousOn.congr_mono
theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s)
    (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s) (h' : Set.EqOn g f s₁) (h₁ : s₁ ⊆ s)
Goal: ContinuousOn g s₁
-- ContinuousOn.congr
theorem ContinuousOn.congr {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : EqOn g f s) :
    ContinuousOn g s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : Set.EqOn g f s)
Goal: ContinuousOn g s
-- continuousOn_congr
theorem continuousOn_congr {f g : α → β} {s : Set α} (h' : EqOn g f s) :
    ContinuousOn g s ↔ ContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s : Set α} (h' : Set.EqOn g f s)
Goal: ContinuousOn g s ↔ ContinuousOn f s
-- ContinuousAt.continuousWithinAt
theorem ContinuousAt.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) :
    ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x)
Goal: ContinuousWithinAt f s x
-- continuousWithinAt_iff_continuousAt
theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) :
    ContinuousWithinAt f s x ↔ ContinuousAt f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x)
Goal: ContinuousWithinAt f s x ↔ ContinuousAt f x
-- ContinuousWithinAt.continuousAt
theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
    (h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x) : ContinuousAt f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x)
Goal: ContinuousAt f x
-- IsOpen.continuousOn_iff
theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
    ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hs : IsOpen s)
Goal: ContinuousOn f s ↔ ∀ ⦃a : α⦄, a ∈ s → ContinuousAt f a
-- ContinuousOn.continuousAt
theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s)
    (hx : s ∈ 𝓝 x) : ContinuousAt f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s) (hx : s ∈ 𝓝 x)
Goal: ContinuousAt f x
-- ContinuousAt.continuousOn
theorem ContinuousAt.continuousOn {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) :
    ContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x)
Goal: ContinuousOn f s
-- ContinuousWithinAt.comp
theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
    (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) :
    ContinuousWithinAt (g ∘ f) s x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α} (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : Set.MapsTo f s t)
Goal: ContinuousWithinAt (g ∘ f) s x
-- ContinuousWithinAt.comp'
theorem ContinuousWithinAt.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
    (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) :
    ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α} (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x
-- ContinuousAt.comp_continuousWithinAt
theorem ContinuousAt.comp_continuousWithinAt {g : β → γ} {f : α → β} {s : Set α} {x : α}
    (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {x : α} (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (g ∘ f) s x
-- ContinuousOn.comp
theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
    (hf : ContinuousOn f s) (h : MapsTo f s t) : ContinuousOn (g ∘ f) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s) (h : Set.MapsTo f s t)
Goal: ContinuousOn (g ∘ f) s
-- ContinuousOn.comp''
@[fun_prop]
theorem ContinuousOn.comp'' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
    (hf : ContinuousOn f s) (h : Set.MapsTo f s t) : ContinuousOn (fun x => g (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s) (h : Set.MapsTo f s t)
Goal: ContinuousOn (fun x => g (f x)) s
-- ContinuousOn.mono
theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) :
    ContinuousOn f t 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s)
Goal: ContinuousOn f t
-- antitone_continuousOn
theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
Goal: Antitone (ContinuousOn f)
-- ContinuousOn.comp'
@[fun_prop]
theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
    (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s)
Goal: ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t)
-- Continuous.continuousOn
@[fun_prop]
theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (h : Continuous f)
Goal: ContinuousOn f s
-- Continuous.continuousWithinAt
theorem Continuous.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : Continuous f) :
    ContinuousWithinAt f s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (h : Continuous f)
Goal: ContinuousWithinAt f s x
-- Continuous.comp_continuousOn
theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g)
    (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s)
Goal: ContinuousOn (g ∘ f) s
-- Continuous.comp_continuousOn'
@[fun_prop]
theorem Continuous.comp_continuousOn'
    {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ}
    {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s) :
    ContinuousOn (fun x ↦ g (f x)) s 
Arguments: {α : Type u_7} {β : Type u_8} {γ : Type u_9} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => g (f x)) s
-- ContinuousOn.comp_continuous
theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s)
    (hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s) (hf : Continuous f) (hs : ∀ (x : α), f x ∈ s)
Goal: Continuous (g ∘ f)
-- continuousOn_apply
@[fun_prop]
theorem continuousOn_apply {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    (i : ι) (s) : ContinuousOn (fun p : ∀ i, π i => p i) s 
Arguments: {ι : Type u_7} {π : ι → Type u_8} [(i : ι) → TopologicalSpace (π i)] (i : ι) (s : Set ((i : ι) → π i))
Goal: ContinuousOn (fun p => p i) s
-- ContinuousWithinAt.preimage_mem_nhdsWithin
theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
    (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x))
Goal: f ⁻¹' t ∈ 𝓝[s] x
-- Set.LeftInvOn.map_nhdsWithin_eq
theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β}
    (h : LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x))
    (hg : ContinuousWithinAt g s x) : map g (𝓝[s] x) = 𝓝[g '' s] g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {g : β → α} {x : β} {s : Set β} (h : Set.LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x)) (hg : ContinuousWithinAt g s x)
Goal: Filter.map g (𝓝[s] x) = 𝓝[g '' s] g x
-- Function.LeftInverse.map_nhds_eq
theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β}
    (h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (range g) (g x))
    (hg : ContinuousAt g x) : map g (𝓝 x) = 𝓝[range g] g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {g : β → α} {x : β} (h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (Set.range g) (g x)) (hg : ContinuousAt g x)
Goal: Filter.map g (𝓝 x) = 𝓝[Set.range g] g x
-- ContinuousWithinAt.preimage_mem_nhdsWithin'
theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β}
    (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x)
Goal: f ⁻¹' t ∈ 𝓝[s] x
-- ContinuousWithinAt.preimage_mem_nhdsWithin''
theorem ContinuousWithinAt.preimage_mem_nhdsWithin''
    {f : α → β} {x : α} {y : β} {s t : Set β}
    (h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x) :
    f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {x : α} {y : β} {s t : Set β} (h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x)
Goal: f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x
-- Filter.EventuallyEq.congr_continuousWithinAt
theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
    (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s : Set α} {x : α} (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x)
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x