Theorems · Theorem · real analysis

intervalIntegral.deriv_integral_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 → deriv (fun u => ∫ (x : ℝ) in a..u, f x) b = f b

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

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

0 results in Mathlib

Foundations

Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace