Twice continuous differentiability under integration

Proved

BianchiIX.contDiffAt_intervalIntegral_two

by trppppp · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Let $E$ be a real normed space, $F:E\times\mathbb R\to\mathbb R$ , $x\in E$ , and $a\leq b$ . Write $\partial_EF(y,t)=dF_{(y,t)}\circ\iota_E$ , where $\iota_E(v)=(v,0)$ , and set

I_F(y)=\int_a^b F(y,t)\,dt.

If $F$ is $C^2$ at $(x,t)$ for every $t\in[a,b]$ , then $I_F$ is $C^2$ at $x$ .

Preamble

import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Topology.Order.Compact
import Mathlib.MeasureTheory.Integral.DominatedConvergence

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX
open Set Filter MeasureTheory
open scoped Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BianchiIX.contDiffAt_intervalIntegral_two {F : E × ℝ → ℝ} {x : E} {a b : ℝ}
    (hab : a ≤ b) (hF : ∀ t ∈ Icc a b, ContDiffAt ℝ 2 F (x, t)) :
    ContDiffAt ℝ 2 (fun y => ∫ t in a..b, F (y, t)) x := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/CompactIntegralSecondRegularity.lean#L12-L28

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