Theorem: ContinuousOn.if'
File: Mathlib/Topology/ContinuousOn.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s : Set α
p : α → Prop
f g : α → β
inst✝ : (a : α) → 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
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s : Set α
p : α → Prop
f g : α → β
inst✝ : (a : α) → 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})
x : α
hx : x ∈ s
⊢ ContinuousWithinAt (fun a => if p a then f a else g a) s x
tactic: intro x hx
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s : Set α
p : α → Prop
f g : α → β
inst✝ : (a : α) → 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})
x : α
hx : x ∈ s
⊢ ContinuousWithinAt (fun a => if p a then f a else g a) s x
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s : Set α
p : α → Prop
f g : α → β
inst✝ : (a : α) → 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})
x : α
hx : x ∈ s
hx' : x ∈ frontier {a | p a}
⊢ ContinuousWithinAt (fun a => if p a then f a else g a) s x

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s : Set α
p : α → Prop
f g : α → β
inst✝ : (a : α) → 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})
x : α
hx : x ∈ s
hx' : x ∉ frontier {a | p a}
⊢ ContinuousWithinAt (fun a => if p a then f a else g a) s x
tactic: by_cases hx' : x ∈ frontier { a | p a }