feat(LipschitzSmooth/Continuity): finite-dim continuity via K-semiconcavity#35
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New file `Mathlib/Analysis/Calculus/LipschitzSmooth/Continuity.lean`. In a
finite-dimensional real inner-product space, a `K`-smooth function is locally
Lipschitz, hence continuous — without any convexity hypothesis on `f` and
without presupposing differentiability.
The proof routes through the **K-semiconcavity decomposition**: the upper-bound
side of the two-sided abs predicate implies `g(x) := f x - K/2 ‖x‖²` is
concave on the whole space (Cannarsa-Sinestrari, *Semiconcave Functions,
Hamilton-Jacobi Equations, and Optimal Control* (2004), Prop. 1.1.3(a)⇔(c)
p.2). Mathlib's `ConcaveOn.locallyLipschitz` (which requires
`[FiniteDimensional ℝ F]`) then gives local Lipschitz of `g`, and adding back
the locally Lipschitz `K/2 ‖x‖²` (via `ContDiff.locallyLipschitz` applied to
`contDiff_norm_sq`) recovers `f`.
The `[FiniteDimensional]` hypothesis is necessary: in infinite-dim, a
discontinuous linear functional `ℓ : F → ℝ` satisfies `LipschitzSmoothWith 0 ℓ`
but fails continuity (Bauschke-Combettes, *Convex Analysis and Monotone
Operator Theory in Hilbert Spaces* (2017), Example 8.42 p.151). The
`[InnerProductSpace]` restriction is for the parallelogram identity used in
the K-semiconcavity decomposition; a general-finite-dim-normed-space refinement
via norm-equivalence transfer is left as a follow-up.
Three theorems:
- `LipschitzSmoothWith.concaveOn_sub_half_sq_norm` — the semiconcavity
decomposition (any dim, Hilbert).
- `LipschitzSmoothWith.locallyLipschitz` — corollary (finite-dim Hilbert).
- `LipschitzSmoothWith.continuous` — corollary (finite-dim Hilbert).
The full `C^{1,1}` characterisation (`Differentiable`, `LipschitzWith
(fderiv ℝ f)`) is deferred to a follow-up; it requires super-gradient
infrastructure not currently in mathlib.
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A
K-smooth function on a finite-dimensional real inner-product space is locally Lipschitz, hence continuous — without any convexity hypothesis onfand without presupposing differentiability. The proof routes through the K-semiconcavity decomposition:g(x) := f x - K/2 ‖x‖²is concave on the whole space (Cannarsa-Sinestrari, Semiconcave Functions (2004), Proposition 1.1.3(a)⇔(c) p.2); mathlib'sConcaveOn.locallyLipschitzthen gives local Lipschitz ofgin finite-dim, and adding back the locally LipschitzK/2 ‖x‖²recoversf.The
[FiniteDimensional]hypothesis is necessary, not convenient — discontinuous linear functionals satisfyLipschitzSmoothWith 0 ℓin infinite-dim (Bauschke-Combettes 2017 Example 8.42 p.151). The[InnerProductSpace]restriction is for the parallelogram identity in the decomposition; a general-finite-dim-normed-space refinement via norm-equivalence transfer is left as a follow-up.The full
C^{1,1}characterisation (Differentiable ℝ f,LipschitzWith K (fderiv ℝ f)) is the natural next step from the K-semiconcavity decomposition exposed here — it goes through the super-gradient sandwich (CS04 Corollary 3.3.8 p.61). Mathlib currently lacks subgradient / superdifferential infrastructure forConcaveOn, so that step is deferred to a separate prerequisite PR introducing super-gradient existence forConcaveOn ℝ univin finite-dim Hilbert via the geometric Hahn-Banach machinery already inMathlib/Analysis/LocallyConvex/Separation.lean. Once that lands, a follow-up PR can extend this file with theDifferentiableandLipschitzWith_fderivresults. Draft because that prereq doesn't exist yet — the content here is internally complete (no sorries).LipschitzSmoothWithandCocoerciveWithleanprover-community/mathlib4#39574Diff for the changes just in this PR over its predecessor: link