Theorems · Definition · global analysis

ContMDiffWithinAt

{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] →
              ModelWithCorners 𝕜 E H →
                {M : Type u_4} →
                  [inst_4 : TopologicalSpace M] →
                    [ChartedSpace H M] →
                      {E' : Type u_5} →
                        [inst_6 : NormedAddCommGroup E'] →
                          [inst_7 : NormedSpace 𝕜 E'] →
                            {H' : Type u_6} →
                              [inst_8 : TopologicalSpace H'] →
                                ModelWithCorners 𝕜 E' H' →
                                  {M' : Type u_7} →
                                    [inst : TopologicalSpace M'] →
                                      [ChartedSpace H' M'] → WithTop ℕ∞ → (M → M') → Set M → M → Prop

ContMDiffWithinAt I I' n f x indicates that the function f : M → M' between manifolds is n times continuously differentiable at x : M within the set s. f is n times continuously differentiable within s at x if it is continuous and it is n times continuously differentiable in this set around x, when read in the preferred chart at x. The parameter n belongs to ℕ∞ω (accessible in the ContDiff scope), i.e. it can be a natural number, ∞, or ω, where C^ω corresponds to analytic functions.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Defs

Cited by

189 results in Mathlib

Foundations

Depth 173 from the axioms, rests on 4,285 definitions · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace