Theorems · Theorem · global analysis

contMDiff_of_tsupport

∀ {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] {E' : Type u_5} [inst_5 : NormedAddCommGroup E'] [inst_6 : NormedSpace 𝕜 E']
  {H' : Type u_6} [inst_7 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7}
  [inst_8 : TopologicalSpace M'] [inst_9 : ChartedSpace H M] [inst_10 : ChartedSpace H' M'] {n : WithTop ℕ∞}
  [inst_11 : Zero M'] {f : M → M'}, (∀ x ∈ tsupport f, ContMDiffAt I I' n f x) → ContMDiff I I' n f

f is continuously differentiable if it is continuously differentiable at each x ∈ tsupport f. See also contMDiff_section_of_tsupport for a similar result for sections of vector bundles.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Basic

Cited by

4 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceChartedSpaceZero