UpperSemicontinuousAt — Mathlib

English

If f is upper semicontinuous at x, then the limsup of f near x does not exceed f(x).

Русский

Если f верхнеполупрерывна в x, то limsup f近 x не превосходит f(x).

LaTeX

$$UpperSemicontinuousAt f x → limsup f (nhds x) ≤ f x$$

Lean4

/-- A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close
enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space,
using an arbitrary `y > f x` instead of `f x + ε`. -/
def UpperSemicontinuousAt (f : α → β) (x : α) :=
  ∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y

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