Theorems · Theorem · real analysis

intervalIntegral.integral_hasDerivAt_right

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ},
  IntervalIntegrable f MeasureTheory.volume a b →
    StronglyMeasurableAtFilter f (nhds b) MeasureTheory.volume →
      ContinuousAt f b → HasDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) b

Fundamental theorem of calculus-1: if f : ℝ → E is integrable on a..b and f is continuous at b, then u ↦ ∫ x in a..u, f x has derivative f b at b.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

2 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace