α : Type u_3
β : Type u_1
γ : Type u_2
δ : Type ?u.336497
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
f : α → β
s : Set β
hg : ContinuousOn g s
hf : ContinuousOn f univ
hs : ∀ (x : α), f x ∈ s
⊢ ContinuousOn (g ∘ f) univ
α : Type u_3
β : Type u_1
γ : Type u_2
δ : Type ?u.336497
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
f : α → β
s : Set β
hg : ContinuousOn g s
hf : Continuous f
hs : ∀ (x : α), f x ∈ s
⊢ Continuous (g ∘ f)
rw [continuous_iff_continuousOn_univ] at *
no goals
α : Type u_3
β : Type u_1
γ : Type u_2
δ : Type ?u.336497
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
g : β → γ
f : α → β
s : Set β
hg : ContinuousOn g s
hf : ContinuousOn f univ
hs : ∀ (x : α), f x ∈ s
⊢ ContinuousOn (g ∘ f) univ
exact hg.comp hf fun x _ => hs x