∀ {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} {s : Set α},
  Topology.IsInducing f → (ContinuousOn g (f '' s) ↔ ContinuousOn (g ∘ f) s)