continuous_piecewise_of_specializes

English

If f is IsInducing, then f x ⤳ f y iff x ⤳ y.

Русский

Если f является индуцирующим, то f x ⤳ f y эквивалентно x ⤳ y.

LaTeX

$$$$ \forall X\ Y, [\text{TopologicalSpace } X], [\text{TopologicalSpace } Y],\ {x,y:X},\ IsInducing f \Rightarrow (f x ⤳ f y \iff x ⤳ y). $$$$

Lean4

theorem continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s) (hf : Continuous f)
    (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) : Continuous (s.piecewise f g) := by
  simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec

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