Theorems · Theorem · approximation theory
Continuous.isBounded_range_iff_isBigO_atTop_of_even
∀ {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] [inst_7 : AddCommGroup β]
[IsOrderedAddMonoid β] {f : β → E},
Continuous f → Function.Even f → (Bornology.IsBounded (Set.range f) ↔ f =O[Filter.atTop] 1)A continuous even function has bounded range if and only if f =O[atTop] 1.
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- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
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- Realstatement · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- Set.rangestatement and proof · 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
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- Asymptotics.IsBigOstatement and proof · cited by 506
- OrderClosedTopologystatement and proof · cited by 445
- NoMaxOrderstatement and proof · cited by 340
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