Theorems · Theorem · approximation theory
Continuous.isBounded_range_iff_isBigO_atTop_atBot
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] {β : Type u_3} [inst_1 : TopologicalSpace β] [inst_2 : LinearOrder β]
[OrderClosedTopology β] [CompactIccSpace β] [NoMaxOrder β] [NoMinOrder β] {f : β → E},
Continuous f → (Bornology.IsBounded (Set.range f) ↔ f =O[Filter.atTop] 1 ∧ f =O[Filter.atBot] 1)A continuous function f on an unbounded linear order with compact intervals has bounded range if
and only if it is O(1) at both atTop and atBot.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- LinearOrderstatement and proof · cited by 8,572
- Filterproof · cited by 8,121
- Set.rangestatement · cited by 4,705
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Continuousstatement and proof · cited by 2,592
- Filter.atTopstatement and proof · cited by 2,405
- Filter.atBotstatement and proof · cited by 512
- Asymptotics.IsBigOstatement and proof · cited by 506
- OrderClosedTopologystatement and proof · cited by 445
- NoMaxOrderstatement and proof · cited by 340
Cited by1
Results whose statement or proof uses this declaration.
- Continuous.isBounded_range_iff_isBigO_atTop_of_evenproof · cited by 0