ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousWithinAt.congr_of_eventuallyEq
theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α}
    (h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    ContinuousWithinAt f₁ s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: ContinuousWithinAt f₁ s x
-- ContinuousWithinAt.congr
theorem ContinuousWithinAt.congr {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
    (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x)
Goal: ContinuousWithinAt f₁ s x
-- ContinuousWithinAt.congr_mono
theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : α}
    (h : ContinuousWithinAt f s x) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) :
    ContinuousWithinAt g s₁ x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s₁ : Set α} {x : α} (h : ContinuousWithinAt f s x) (h' : Set.EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x)
Goal: ContinuousWithinAt g s₁ x
-- continuousOn_const
@[fun_prop]
theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun _ => c) s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {c : β}
Goal: ContinuousOn (fun x => c) s
-- continuousWithinAt_const
theorem continuousWithinAt_const {b : β} {s : Set α} {x : α} :
    ContinuousWithinAt (fun _ : α => b) s x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {b : β} {s : Set α} {x : α}
Goal: ContinuousWithinAt (fun x => b) s x
-- continuousOn_id
theorem continuousOn_id {s : Set α} : ContinuousOn id s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α}
Goal: ContinuousOn id s
-- continuousOn_id'
@[fun_prop]
theorem continuousOn_id' (s : Set α) : ContinuousOn (fun x : α => x) s 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α)
Goal: ContinuousOn (fun x => x) s
-- continuousWithinAt_id
theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {x : α}
Goal: ContinuousWithinAt id s x
-- continuousOn_open_iff
theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
    ContinuousOn f s ↔ ∀ t, IsOpen t → IsOpen (s ∩ f ⁻¹' t) 
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 ↔ ∀ (t : Set β), IsOpen t → IsOpen (s ∩ f ⁻¹' t)
-- ContinuousOn.isOpen_inter_preimage
theorem ContinuousOn.isOpen_inter_preimage {f : α → β} {s : Set α} {t : Set β}
    (hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' 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) (hs : IsOpen s) (ht : IsOpen t)
Goal: IsOpen (s ∩ f ⁻¹' t)
-- ContinuousOn.isOpen_preimage
theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
    (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s) (hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t)
Goal: IsOpen (f ⁻¹' t)
-- ContinuousOn.preimage_isClosed_of_isClosed
theorem ContinuousOn.preimage_isClosed_of_isClosed {f : α → β} {s : Set α} {t : Set β}
    (hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' 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) (hs : IsClosed s) (ht : IsClosed t)
Goal: IsClosed (s ∩ f ⁻¹' t)
-- ContinuousOn.preimage_interior_subset_interior_preimage
theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
    (hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' 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) (hs : IsOpen s)
Goal: s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t)
-- continuousOn_of_locally_continuousOn
theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
    (h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t)) : ContinuousOn f s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t))
Goal: ContinuousOn f s
-- continuousOn_to_generateFrom_iff
theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α → β} :
    @ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {s : Set α} {T : Set (Set β)} {f : α → β}
Goal: ContinuousOn f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x
-- continuousOn_isOpen_of_generateFrom
theorem continuousOn_isOpen_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
    (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
    @ContinuousOn α β _ (.generateFrom T) f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_7} {s : Set α} {T : Set (Set β)} {f : α → β} (h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t))
Goal: ContinuousOn f s
-- ContinuousWithinAt.prod
theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}
    (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun x => (f x, g x)) s x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : α → γ} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (fun x => (f x, g x)) s x
-- ContinuousOn.prod
@[fun_prop]
theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) : ContinuousOn (fun x => (f x, g x)) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => (f x, g x)) s
-- ContinuousAt.comp₂_continuousWithinAt
theorem ContinuousAt.comp₂_continuousWithinAt {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α}
    {s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x)
    (hh : ContinuousWithinAt h s x) :
    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x 
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 : β × γ → δ} {g : α → β} {h : α → γ} {x : α} {s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x)
Goal: ContinuousWithinAt (fun x => f (g x, h x)) s x
-- ContinuousAt.comp₂_continuousWithinAt_of_eq
theorem ContinuousAt.comp₂_continuousWithinAt_of_eq {f : β × γ → δ} {g : α → β}
    {h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y)
    (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y) :
    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x 
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 : β × γ → δ} {g : α → β} {h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y) (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y)
Goal: ContinuousWithinAt (fun x => f (g x, h x)) s x
-- Inducing.continuousWithinAt_iff
theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
    {x : α} : 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] {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α} {x : α}
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x
-- Inducing.continuousOn_iff
theorem Inducing.continuousOn_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α} :
    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] {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
Goal: ContinuousOn f s ↔ ContinuousOn (g ∘ f) s
-- Embedding.continuousOn_iff
theorem Embedding.continuousOn_iff {f : α → β} {g : β → γ} (hg : Embedding g) {s : Set α} :
    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] {f : α → β} {g : β → γ} (hg : Embedding g) {s : Set α}
Goal: ContinuousOn f s ↔ ContinuousOn (g ∘ f) s
-- Embedding.map_nhdsWithin_eq
theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set α) (x : α) :
    map f (𝓝[s] x) = 𝓝[f '' s] f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Embedding f) (s : Set α) (x : α)
Goal: Filter.map f (𝓝[s] x) = 𝓝[f '' s] f x
-- OpenEmbedding.map_nhdsWithin_preimage_eq
theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbedding f) (s : Set β)
    (x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) (s : Set β) (x : α)
Goal: Filter.map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x
-- QuotientMap.continuousOn_isOpen_iff
theorem QuotientMap.continuousOn_isOpen_iff {f : α → β} {g : β → γ} (h : QuotientMap f) {s : Set β}
    (hs : IsOpen s) : ContinuousOn g s ↔ ContinuousOn (g ∘ f) (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : β → γ} (h : QuotientMap f) {s : Set β} (hs : IsOpen s)
Goal: ContinuousOn g s ↔ ContinuousOn (g ∘ f) (f ⁻¹' s)
-- continuousWithinAt_of_not_mem_closure
theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} (hx : x ∉ closure s) :
    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} (hx : x ∉ closure s)
Goal: ContinuousWithinAt f s x
-- ContinuousOn.if'
theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
    (hpf : ∀ a ∈ s ∩ frontier { a | p a },
      Tendsto f (𝓝[s ∩ { a | p a }] a) (𝓝 <| if p a then f a else g a))
    (hpg :
      ∀ a ∈ s ∩ frontier { a | p a },
        Tendsto g (𝓝[s ∩ { a | ¬p a }] a) (𝓝 <| if p a then f a else g a))
    (hf : ContinuousOn f <| s ∩ { a | p a }) (hg : ContinuousOn g <| s ∩ { a | ¬p a }) :
    ContinuousOn (fun a => if p a then f a else g a) s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {p : α → Prop} {f g : α → β} [(a : α) → Decidable (p a)] (hpf : ∀ a ∈ s ∩ frontier {a | p a}, Filter.Tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 (if p a then f a else g a))) (hpg : ∀ a ∈ s ∩ frontier {a | p a}, Filter.Tendsto g (𝓝[s ∩ {a | ¬p a}] a) (𝓝 (if p a then f a else g a))) (hf : ContinuousOn f (s ∩ {a | p a})) (hg : ContinuousOn g (s ∩ {a | ¬p a}))
Goal: ContinuousOn (fun a => if p a then f a else g a) s
-- ContinuousOn.piecewise'
theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
    (hpf : ∀ a ∈ s ∩ frontier t, Tendsto f (𝓝[s ∩ t] a) (𝓝 (piecewise t f g a)))
    (hpg : ∀ a ∈ s ∩ frontier t, Tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (piecewise t f g a)))
    (hf : ContinuousOn f <| s ∩ t) (hg : ContinuousOn g <| s ∩ tᶜ) :
    ContinuousOn (piecewise t f g) s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s t : Set α} {f g : α → β} [(a : α) → Decidable (a ∈ t)] (hpf : ∀ a ∈ s ∩ frontier t, Filter.Tendsto f (𝓝[s ∩ t] a) (𝓝 (t.piecewise f g a))) (hpg : ∀ a ∈ s ∩ frontier t, Filter.Tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (t.piecewise f g a))) (hf : ContinuousOn f (s ∩ t)) (hg : ContinuousOn g (s ∩ tᶜ))
Goal: ContinuousOn (t.piecewise f g) s
-- ContinuousOn.if
theorem ContinuousOn.if {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
    [∀ a, Decidable (p a)] {s : Set α} {f g : α → β}
    (hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
    (hf : ContinuousOn f <| s ∩ closure { a | p a })
    (hg : ContinuousOn g <| s ∩ closure { a | ¬p a }) :
    ContinuousOn (fun a => if p a then f a else g a) s 
Arguments: {α : Type u_7} {β : Type u_8} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop} [(a : α) → Decidable (p a)] {s : Set α} {f g : α → β} (hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a) (hf : ContinuousOn f (s ∩ closure {a | p a})) (hg : ContinuousOn g (s ∩ closure {a | ¬p a}))
Goal: ContinuousOn (fun a => if p a then f a else g a) s
-- ContinuousOn.piecewise
theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
    (ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f <| s ∩ closure t)
    (hg : ContinuousOn g <| s ∩ closure tᶜ) : ContinuousOn (piecewise t f g) s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s t : Set α} {f g : α → β} [(a : α) → Decidable (a ∈ t)] (ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f (s ∩ closure t)) (hg : ContinuousOn g (s ∩ closure tᶜ))
Goal: ContinuousOn (t.piecewise f g) s
-- continuous_if'
theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
    (hpf : ∀ a ∈ frontier { x | p x }, Tendsto f (𝓝[{ x | p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
    (hpg : ∀ a ∈ frontier { x | p x }, Tendsto g (𝓝[{ x | ¬p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
    (hf : ContinuousOn f { x | p x }) (hg : ContinuousOn g { x | ¬p x }) :
    Continuous fun a => ite (p a) (f a) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop} {f g : α → β} [(a : α) → Decidable (p a)] (hpf : ∀ a ∈ frontier {x | p x}, Filter.Tendsto f (𝓝[{x | p x}] a) (𝓝 (if p a then f a else g a))) (hpg : ∀ a ∈ frontier {x | p x}, Filter.Tendsto g (𝓝[{x | ¬p x}] a) (𝓝 (if p a then f a else g a))) (hf : ContinuousOn f {x | p x}) (hg : ContinuousOn g {x | ¬p x})
Goal: Continuous fun a => if p a then f a else g a
-- continuous_if
theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
    (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : ContinuousOn f (closure { x | p x }))
    (hg : ContinuousOn g (closure { x | ¬p x })) :
    Continuous fun a => if p a then f a else g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop} {f g : α → β} [(a : α) → Decidable (p a)] (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : ContinuousOn f (closure {x | p x})) (hg : ContinuousOn g (closure {x | ¬p x}))
Goal: Continuous fun a => if p a then f a else g a
-- Continuous.if
theorem Continuous.if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
    (hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : Continuous f) (hg : Continuous g) :
    Continuous fun a => if p a then f a else g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop} {f g : α → β} [(a : α) → Decidable (p a)] (hp : ∀ a ∈ frontier {x | p x}, f a = g a) (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun a => if p a then f a else g a
-- continuous_if_const
theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
    (hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f) (hg : ¬p → Continuous g)
Goal: Continuous fun a => if p then f a else g a
-- Continuous.if_const
theorem Continuous.if_const (p : Prop) {f g : α → β} [Decidable p] (hf : Continuous f)
    (hg : Continuous g) : Continuous fun a => if p then f a else g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (p : Prop) {f g : α → β} [Decidable p] (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun a => if p then f a else g a
-- continuous_piecewise
theorem continuous_piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
    (hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s))
    (hg : ContinuousOn g (closure sᶜ)) : Continuous (piecewise s f g) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f g : α → β} [(a : α) → Decidable (a ∈ s)] (hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure sᶜ))
Goal: Continuous (s.piecewise f g)
-- Continuous.piecewise
theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
    (hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g) :
    Continuous (piecewise s f g) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f g : α → β} [(a : α) → Decidable (a ∈ s)] (hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g)
Goal: Continuous (s.piecewise f g)
-- continuous_indicator
@[to_additive]
lemma continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :
    Continuous (mulIndicator s f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f : α → β} {s : Set α} (hs : ∀ a ∈ frontier s, f a = 0) (hf : ContinuousOn f (closure s))
Goal: Continuous (s.indicator f)
-- continuous_mulIndicator
@[to_additive]
lemma continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :
    Continuous (mulIndicator s f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [One β] {f : α → β} {s : Set α} (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s))
Goal: Continuous (s.mulIndicator f)
-- Continuous.mulIndicator
@[to_additive]
protected lemma Continuous.mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : Continuous f) :
    Continuous (mulIndicator s f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [One β] {f : α → β} {s : Set α} (hs : ∀ a ∈ frontier s, f a = 1) (hf : Continuous f)
Goal: Continuous (s.mulIndicator f)
-- Continuous.indicator
@[to_additive]
protected lemma Continuous.mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : Continuous f) :
    Continuous (mulIndicator s f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f : α → β} {s : Set α} (hs : ∀ a ∈ frontier s, f a = 0) (hf : Continuous f)
Goal: Continuous (s.indicator f)
-- IsOpen.ite'
theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
    (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') 
Arguments: {α : Type u_1} [TopologicalSpace α] {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s') (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s')
Goal: IsOpen (t.ite s s')
-- IsOpen.ite
theorem IsOpen.ite {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
    (ht : s ∩ frontier t = s' ∩ frontier t) : IsOpen (t.ite s s') 
Arguments: {α : Type u_1} [TopologicalSpace α] {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s') (ht : s ∩ frontier t = s' ∩ frontier t)
Goal: IsOpen (t.ite s s')
-- ite_inter_closure_eq_of_inter_frontier_eq
theorem ite_inter_closure_eq_of_inter_frontier_eq {s s' t : Set α}
    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure t = s ∩ closure t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s s' t : Set α} (ht : s ∩ frontier t = s' ∩ frontier t)
Goal: t.ite s s' ∩ closure t = s ∩ closure t
-- ite_inter_closure_compl_eq_of_inter_frontier_eq
theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
    (ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure tᶜ = s' ∩ closure tᶜ 
Arguments: {α : Type u_1} [TopologicalSpace α] {s s' t : Set α} (ht : s ∩ frontier t = s' ∩ frontier t)
Goal: t.ite s s' ∩ closure tᶜ = s' ∩ closure tᶜ
-- continuousOn_piecewise_ite'
theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
    (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure tᶜ))
    (H : s ∩ frontier t = s' ∩ frontier t) (Heq : EqOn f f' (s ∩ frontier t)) :
    ContinuousOn (t.piecewise f f') (t.ite s s') 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s s' t : Set α} {f f' : α → β} [(x : α) → Decidable (x ∈ t)] (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure tᶜ)) (H : s ∩ frontier t = s' ∩ frontier t) (Heq : Set.EqOn f f' (s ∩ frontier t))
Goal: ContinuousOn (t.piecewise f f') (t.ite s s')
-- continuousOn_piecewise_ite
theorem continuousOn_piecewise_ite {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
    (h : ContinuousOn f s) (h' : ContinuousOn f' s') (H : s ∩ frontier t = s' ∩ frontier t)
    (Heq : EqOn f f' (s ∩ frontier t)) : ContinuousOn (t.piecewise f f') (t.ite s s') 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s s' t : Set α} {f f' : α → β} [(x : α) → Decidable (x ∈ t)] (h : ContinuousOn f s) (h' : ContinuousOn f' s') (H : s ∩ frontier t = s' ∩ frontier t) (Heq : Set.EqOn f f' (s ∩ frontier t))
Goal: ContinuousOn (t.piecewise f f') (t.ite s s')
-- frontier_inter_open_inter
theorem frontier_inter_open_inter {s t : Set α} (ht : IsOpen t) :
    frontier (s ∩ t) ∩ t = frontier s ∩ t 
Arguments: {α : Type u_1} [TopologicalSpace α] {s t : Set α} (ht : IsOpen t)
Goal: frontier (s ∩ t) ∩ t = frontier s ∩ t
-- continuousOn_fst
theorem continuousOn_fst {s : Set (α × β)} : ContinuousOn Prod.fst s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α × β)}
Goal: ContinuousOn Prod.fst s
-- continuousWithinAt_fst
theorem continuousWithinAt_fst {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.fst s p 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α × β)} {p : α × β}
Goal: ContinuousWithinAt Prod.fst s p
-- ContinuousOn.fst
@[fun_prop]
theorem ContinuousOn.fst {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
    ContinuousOn (fun x => (f x).1) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => (f x).1) s
-- ContinuousWithinAt.fst
theorem ContinuousWithinAt.fst {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => (f x).fst) s a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β × γ} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => (f x).1) s a
-- continuousOn_snd
theorem continuousOn_snd {s : Set (α × β)} : ContinuousOn Prod.snd s 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α × β)}
Goal: ContinuousOn Prod.snd s
-- continuousWithinAt_snd
theorem continuousWithinAt_snd {s : Set (α × β)} {p : α × β} : ContinuousWithinAt Prod.snd s p 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set (α × β)} {p : α × β}
Goal: ContinuousWithinAt Prod.snd s p
-- ContinuousOn.snd
@[fun_prop]
theorem ContinuousOn.snd {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s) :
    ContinuousOn (fun x => (f x).2) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β × γ} {s : Set α} (hf : ContinuousOn f s)
Goal: ContinuousOn (fun x => (f x).2) s