∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β]
  [inst_2 : TopologicalSpace γ] {f : α → β} {g : β → γ},
  Topology.IsQuotientMap f → ∀ {s : Set β}, IsOpen s → (ContinuousOn g s ↔ ContinuousOn (g ∘ f) (f ⁻¹' s))