Smoothness of Real.sqrt

In this file we prove that Real.sqrt is infinitely smooth at all points x ≠ 0 and provide some dot-notation lemmas.

Tags

sqrt, differentiable

noncomputable def Real.sqPartialHomeomorph : OpenPartialHomeomorph ℝ ℝ

Local homeomorph between (0, +∞) and (0, +∞) with toFun = (· ^ 2) and invFun = Real.sqrt.

theorem Real.deriv_sqrt_aux {x : ℝ} (hx : x ≠ 0) : HasStrictDerivAt (fun (x : ℝ) => √x) (1 / (2 * √x)) x ∧ ∀ (n : WithTop ℕ∞), ContDiffAt ℝ n (fun (x : ℝ) => √x) x

theorem Real.hasStrictDerivAt_sqrt {x : ℝ} (hx : x ≠ 0) : HasStrictDerivAt (fun (x : ℝ) => √x) (1 / (2 * √x)) x

theorem Real.contDiffAt_sqrt {x : ℝ} {n : WithTop ℕ∞} (hx : x ≠ 0) : ContDiffAt ℝ n (fun (x : ℝ) => √x) x

theorem Real.hasDerivAt_sqrt {x : ℝ} (hx : x ≠ 0) : HasDerivAt (fun (x : ℝ) => √x) (1 / (2 * √x)) x

theorem HasDerivWithinAt.sqrt {f : ℝ → ℝ} {s : Set ℝ} {f' x : ℝ} (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0) : HasDerivWithinAt (fun (y : ℝ) => √(f y)) (f' / (2 * √(f x))) s x

theorem HasDerivAt.sqrt {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun (y : ℝ) => √(f y)) (f' / (2 * √(f x))) x

theorem HasStrictDerivAt.sqrt {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) : HasStrictDerivAt (fun (t : ℝ) => √(f t)) (f' / (2 * √(f x))) x

theorem derivWithin_sqrt {f : ℝ → ℝ} {s : Set ℝ} {x : ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0) (hxs : UniqueDiffWithinAt ℝ s x) : derivWithin (fun (x : ℝ) => √(f x)) s x = derivWithin f s x / (2 * √(f x))

@[simp]

theorem deriv_sqrt {f : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) : deriv (fun (x : ℝ) => √(f x)) x = deriv f x / (2 * √(f x))

theorem HasFDerivAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {f' : StrongDual ℝ E} (hf : HasFDerivAt f f' x) (hx : f x ≠ 0) : HasFDerivAt (fun (y : E) => √(f y)) ((1 / (2 * √(f x))) • f') x

theorem HasStrictFDerivAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {f' : StrongDual ℝ E} (hf : HasStrictFDerivAt f f' x) (hx : f x ≠ 0) : HasStrictFDerivAt (fun (y : E) => √(f y)) ((1 / (2 * √(f x))) • f') x

theorem HasFDerivWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {x : E} {f' : StrongDual ℝ E} (hf : HasFDerivWithinAt f f' s x) (hx : f x ≠ 0) : HasFDerivWithinAt (fun (y : E) => √(f y)) ((1 / (2 * √(f x))) • f') s x

theorem DifferentiableWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {x : E} (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0) : DifferentiableWithinAt ℝ (fun (y : E) => √(f y)) s x

theorem DifferentiableAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) : DifferentiableAt ℝ (fun (y : E) => √(f y)) x

theorem DifferentiableOn.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hf : DifferentiableOn ℝ f s) (hs : ∀ x ∈ s, f x ≠ 0) : DifferentiableOn ℝ (fun (y : E) => √(f y)) s

theorem Differentiable.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hf : Differentiable ℝ f) (hs : ∀ (x : E), f x ≠ 0) : Differentiable ℝ fun (y : E) => √(f y)

theorem fderivWithin_sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {x : E} (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0) (hxs : UniqueDiffWithinAt ℝ s x) : fderivWithin ℝ (fun (x : E) => √(f x)) s x = (1 / (2 * √(f x))) • fderivWithin ℝ f s x

@[simp]

theorem fderiv_sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) : fderiv ℝ (fun (x : E) => √(f x)) x = (1 / (2 * √(f x))) • fderiv ℝ f x

theorem ContDiffAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} {x : E} (hf : ContDiffAt ℝ n f x) (hx : f x ≠ 0) : ContDiffAt ℝ n (fun (y : E) => √(f y)) x

theorem ContDiffWithinAt.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} {s : Set E} {x : E} (hf : ContDiffWithinAt ℝ n f s x) (hx : f x ≠ 0) : ContDiffWithinAt ℝ n (fun (y : E) => √(f y)) s x

theorem ContDiffOn.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} {s : Set E} (hf : ContDiffOn ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0) : ContDiffOn ℝ n (fun (y : E) => √(f y)) s

theorem ContDiff.sqrt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : WithTop ℕ∞} (hf : ContDiff ℝ n f) (h : ∀ (x : E), f x ≠ 0) : ContDiff ℝ n fun (y : E) => √(f y)