English
A family of functions F on α → β → γ with sufficiently nice conditions has TendstoUniformly: for each x, TendstoUniformly f (f x) (nhds x).
Русский
Familie функций F на α → β → γ удовлетворяет условиям, тогда существует TendstoUniformly: для каждого x имеет предел в nhds x.
LaTeX
$$\\text{TendstoUniformly}(f)\\to f(x) \\, (nhds x)$$
Lean4
/-- A continuous family of functions `α → β → γ` tends uniformly to its value at `x`
if `α` is weakly locally compact and `β` is compact. -/
theorem tendstoUniformly [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ] (f : α → β → γ)
(h : Continuous ↿f) (x : α) : TendstoUniformly f (f x) (𝓝 x) :=
let ⟨K, hK, hxK⟩ := exists_compact_mem_nhds x
have : UniformContinuousOn ↿f (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ) h.continuousOn
this.tendstoUniformly hxK
