Theorems · Theorem · real analysis

Continuous.integral_hasStrictDerivAt

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

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

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

NormedAddCommGroupNormedSpaceCompleteSpace