∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : UniformSpace α] [inst_1 : UniformSpace β]
  [inst_2 : UniformSpace γ] {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ} [T0Space γ]
  {g : α → γ}, (∀ (b : β), g (e b) = f b) → Continuous g → ⋯.extend f = g