Theorems · Theorem · global analysis

contMDiffWithinAt_iff_contMDiffWithinAt_nhdsWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_5} [inst_6 : NormedAddCommGroup E']
  [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {s : Set M} {x : M}
  {n : WithTop ℕ∞} [IsManifold I n M] [IsManifold I' n M'],
  n ≠ ↑⊤ → (ContMDiffWithinAt I I' n f s x ↔ ∀ᶠ (x' : M) in nhdsWithin x (insert x s), ContMDiffWithinAt I I' n f s x')

Note: This does not hold for n = ∞. f being C^∞ at x means that for every n, f is C^n on some neighborhood of x, but this neighborhood can depend on n.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Defs

Cited by

3 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold