Theorems · Theorem · real analysis

Continuous.deriv_integral

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] (f : ℝ → E),
  Continuous f → ∀ (a b : ℝ), deriv (fun u => ∫ (x : ℝ) in a..u, f x) b = f b

Fundamental theorem of calculus-1, derivative in the right endpoint. If f : ℝ → E is continuous, then the derivative of u ↦ ∫ x in a..u, f x at b is 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