Theorems · Theorem · real analysis

intervalIntegral.integral_hasDerivWithinAt_left

∀ {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 →
    ∀ {s t : Set ℝ} [intervalIntegral.FTCFilter a (nhdsWithin a s) (nhdsWithin a t)],
      StronglyMeasurableAtFilter f (nhdsWithin a t) MeasureTheory.volume →
        ContinuousWithinAt f t a → HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) s a

Fundamental theorem of calculus: if f : ℝ → E is integrable on a..b and f x is continuous from the left or from the right at a, then u ↦ ∫ x in u..b, f x has left (resp., right) derivative -f a at a.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceintervalIntegral.FTCFilter