comp_lowerSemicontinuous_antitone

English

If g is continuous at x and f is lower semicontinuous, then the composed function g∘f is lower semicontinuous at x.

Русский

Если g непрерывна в x и f нижне полупрерывна, то g∘f непрерывна в x по нижней полупрерывности.

LaTeX

$$$\\text{LowerSemicontinuousAt } f x \\land \\text{ContinuousAt } g x \\Rightarrow \\text{LowerSemicontinuousAt } (g∘f) x$$$

Lean4

theorem comp_lowerSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuous f)
    (gmon : Antitone g) : UpperSemicontinuous (g ∘ f) := fun x =>
  hg.continuousAt.comp_lowerSemicontinuousAt_antitone (hf x) gmon

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