lowerSemicontinuousAt_indicator — Mathlib

English

If s is closed and y ≤ 0, then the indicator function of s is lower semicontinuous at any x.

Русский

Если s замкнуто, и y ≤ 0, то индикатор s имеет нижнюю полупрерывность в любой точке x.

LaTeX

$$$\operatorname{LowerSemicontinuousAt}(\mathbf{1}_{s} \cdot y, x)$$$

Lean4

theorem lowerSemicontinuousAt_indicator (hs : IsClosed s) (hy : y ≤ 0) :
    LowerSemicontinuousAt (indicator s fun _x => y) x :=
  (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x

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