continuousAt_lift₂ — Mathlib

English

ContinuousAt (lift₂ f hf) (mk x, mk y) is equivalent to ContinuousAt (uncurry f) (x, y).

Русский

НепрерывностьAt (lift₂ f hf) в точке (mk x, mk y) эквивалентна непрерывности At (f) в точке (x, y).

LaTeX

$$ContinuousAt (Function.uncurry (SeparationQuotient.lift₂ f hf)) (Prod.mk x y) ↔ ContinuousAt (Function.uncurry f) (x, y)$$

Lean4

@[simp]
theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y} :
    ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
  tendsto_lift₂_nhds

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