comp_continuousAt — Mathlib

English

If hg is continuous at f(x), hf is lower semicontinuous at x, and g is antitone, then g∘f is upper semicontinuous at x.

Русский

Если hg непрерывна в f(x), hf — нижняя полупрерывность в x, и g устойчива к антимонотонности, тогда g∘f — верхняя полупрерывность в x.

LaTeX

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

Lean4

theorem comp_continuousAt {f : α → β} {g : ι → α} {x : ι} (hf : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
    LowerSemicontinuousAt (fun x ↦ f (g x)) x := fun _ lt ↦ hg.eventually (hf _ lt)

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