Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Exact Lean statement
lemma continuousWithinAt_Iio_indicator_Ioc {α M : Type*} [LinearOrder α] [TopologicalSpace α]
[ClosedIicTopology α] [Zero M] [TopologicalSpace M] (a b : α) (c : M) (t : α) :
ContinuousWithinAt ((Set.Ioc a b).indicator (fun _ ↦ c)) (Set.Iio t) tFormal artifact
Lean source
lemma continuousWithinAt_Iio_indicator_Ioc {α M : Type*} [LinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] [Zero M] [TopologicalSpace M] (a b : α) (c : M) (t : α) : ContinuousWithinAt ((Set.Ioc a b).indicator (fun _ ↦ c)) (Set.Iio t) t := by refine continuousWithinAt_const.congr_of_eventuallyEq ?_ rfl rcases le_or_gt t a with hta | hat · filter_upwards [self_mem_nhdsWithin] with x hx simp_all [hx.le.trans hta] · rcases le_or_gt t b with htb | hbt · filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds hat)] with x hx_lt hx_gt simp_all [hx_lt.le.trans htb] · filter_upwards [mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds hbt)] with x hx_gt simp_all- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:1125-1137
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Related declarations
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.
Dense monotone of is Right Continuous
Dense.monotone_of_isRightContinuous
Project documentation
If f is monotone on a dense set D and is right continuous, then f is monotone. We prove under the assumption that α has a top element ⊤ and ⊤ ∈ D, which is a necessary assumption because otherwise it is possible that ⊤ is an isolated point. This theorem should be also true when α satisfies NoTopOrder α.
Source project: Brownian motion
Person-level attribution pending.