leanprover-community/mathlib4

Prove that inversion is discontinuous at the center

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#5,939 opened on Jul 16, 2023

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Description

  • Prove Tendsto (inversion c R) (𝓝[≠] c) cobounded if R ≠ 0.
  • Deduce that inversion c R is discontinuous at c if R ≠ 0 and the vector space is nontrivial.
  • Deduce that fderiv (inversion c R) c is zero, thus fderiv (inversion c R) x = ((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x for all x (see #5937 for the case x ≠ c).

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Research direction
Study existing definitions of inversion and reflection in mathlib4. Examine PR #5937 for the case when x ≠ c. Prove the first bullet point by showing that inversion tends to cobounded. Then deduce discontinuity and compute the Fréchet derivative.
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backend
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3
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1-2 days
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Clear
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GitLean basics
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35