English
If f is strictly monotone on a right neighborhood and its image meets every rightward interval, then f is right-continuous at a.
Русский
Если f строго монотонна на правой окрестности и образ пересекает каждое правое интервал, то f сходится слева направо в a.
LaTeX
$$$\text{StrictMonoOn}(f, s) \Rightarrow s \in \mathcal{N}[\ge] a \Rightarrow \forall b>f(a), \exists c\in s, f(c) \in \mathrm{Ioc}(f(a), b) \Rightarrow \text{ContinuousWithinAt}(f, \mathrm{Ici}(a), a)$$$
Lean4
/-- An antitone map has a limit to the left of any point `x`, equal to `sInf (f '' (Iio x))`. -/
theorem tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Af : Antitone f) (x : α) :
Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) :=
Monotone.tendsto_nhdsLT Af.dual_right x
