α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ (fun x => -1) =ᶠ[nhds a] ↑SignType.sign
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ ContinuousAt (↑SignType.sign) a
refine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ ∃ t, (∀ (x : α), x ∈ t → -1 = ↑SignType.sign x) ∧ IsOpen t ∧ a ∈ t
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ (fun x => -1) =ᶠ[nhds a] ↑SignType.sign
rw [Filter.EventuallyEq, eventually_nhds_iff]
no goals
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ ∃ t, (∀ (x : α), x ∈ t → -1 = ↑SignType.sign x) ∧ IsOpen t ∧ a ∈ t
exact ⟨{ x | x < 0 }, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩