comp_lowerSemicontinuousOn_antitone

English

If g is continuous and f is lower semicontinuous, then their composition is lower semicontinuous.

Русский

Если g непрерывна и f — нижне полупрерывна, то их композиция сохраняет нижнюю полупрерывность.

LaTeX

$$$\\text{LowerSemicontinuous } f \\;\\land\\; \\text{Continuous } g \\Rightarrow \\text{LowerSemicontinuous } (f∘g)$$$

Lean4

theorem comp_lowerSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuousOn f s)
    (gmon : Antitone g) : UpperSemicontinuousOn (g ∘ f) s := fun x hx =>
  hg.continuousAt.comp_lowerSemicontinuousWithinAt_antitone (hf x hx) gmon

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