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Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Continuity of power functions #

This file contains lemmas about continuity of the power functions on , , ℝ≥0, and ℝ≥0∞.

Continuity for complex powers #

theorem zero_cpow_eq_nhds {b : } (hb : b 0) :
(fun (x : ) => 0 ^ x) =ᶠ[nhds b] 0
theorem cpow_eq_nhds {a : } {b : } (ha : a 0) :
(fun (x : ) => x ^ b) =ᶠ[nhds a] fun (x : ) => Complex.exp (Complex.log x * b)
theorem cpow_eq_nhds' {p : × } (hp_fst : p.1 0) :
(fun (x : × ) => x.1 ^ x.2) =ᶠ[nhds p] fun (x : × ) => Complex.exp (Complex.log x.1 * x.2)
theorem continuousAt_const_cpow {a : } {b : } (ha : a 0) :
ContinuousAt (fun (x : ) => a ^ x) b
theorem continuousAt_const_cpow' {a : } {b : } (h : b 0) :
ContinuousAt (fun (x : ) => a ^ x) b
theorem continuousAt_cpow {p : × } (hp_fst : p.1 Complex.slitPlane) :
ContinuousAt (fun (x : × ) => x.1 ^ x.2) p

The function z ^ w is continuous in (z, w) provided that z does not belong to the interval (-∞, 0] on the real line. See also Complex.continuousAt_cpow_zero_of_re_pos for a version that works for z = 0 but assumes 0 < re w.

theorem continuousAt_cpow_const {a : } {b : } (ha : a Complex.slitPlane) :
ContinuousAt (fun (x : ) => Complex.cpow x b) a
theorem Filter.Tendsto.cpow {α : Type u_1} {l : Filter α} {f : α} {g : α} {a : } {b : } (hf : Filter.Tendsto f l (nhds a)) (hg : Filter.Tendsto g l (nhds b)) (ha : a Complex.slitPlane) :
Filter.Tendsto (fun (x : α) => f x ^ g x) l (nhds (a ^ b))
theorem Filter.Tendsto.const_cpow {α : Type u_1} {l : Filter α} {f : α} {a : } {b : } (hf : Filter.Tendsto f l (nhds b)) (h : a 0 b 0) :
Filter.Tendsto (fun (x : α) => a ^ f x) l (nhds (a ^ b))
theorem ContinuousWithinAt.cpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {s : Set α} {a : α} (hf : ContinuousWithinAt f s a) (hg : ContinuousWithinAt g s a) (h0 : f a Complex.slitPlane) :
ContinuousWithinAt (fun (x : α) => f x ^ g x) s a
theorem ContinuousWithinAt.const_cpow {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {a : α} {b : } (hf : ContinuousWithinAt f s a) (h : b 0 f a 0) :
ContinuousWithinAt (fun (x : α) => b ^ f x) s a
theorem ContinuousAt.cpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {a : α} (hf : ContinuousAt f a) (hg : ContinuousAt g a) (h0 : f a Complex.slitPlane) :
ContinuousAt (fun (x : α) => f x ^ g x) a
theorem ContinuousAt.const_cpow {α : Type u_1} [TopologicalSpace α] {f : α} {a : α} {b : } (hf : ContinuousAt f a) (h : b 0 f a 0) :
ContinuousAt (fun (x : α) => b ^ f x) a
theorem ContinuousOn.cpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h0 : as, f a Complex.slitPlane) :
ContinuousOn (fun (x : α) => f x ^ g x) s
theorem ContinuousOn.const_cpow {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {b : } (hf : ContinuousOn f s) (h : b 0 as, f a 0) :
ContinuousOn (fun (x : α) => b ^ f x) s
theorem Continuous.cpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} (hf : Continuous f) (hg : Continuous g) (h0 : ∀ (a : α), f a Complex.slitPlane) :
Continuous fun (x : α) => f x ^ g x
theorem Continuous.const_cpow {α : Type u_1} [TopologicalSpace α] {f : α} {b : } (hf : Continuous f) (h : b 0 ∀ (a : α), f a 0) :
Continuous fun (x : α) => b ^ f x
theorem ContinuousOn.cpow_const {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {b : } (hf : ContinuousOn f s) (h : as, f a Complex.slitPlane) :
ContinuousOn (fun (x : α) => f x ^ b) s

Continuity for real powers #

theorem Real.rpow_eq_nhds_of_neg {p : × } (hp_fst : p.1 < 0) :
(fun (x : × ) => x.1 ^ x.2) =ᶠ[nhds p] fun (x : × ) => Real.exp (Real.log x.1 * x.2) * Real.cos (x.2 * Real.pi)
theorem Real.rpow_eq_nhds_of_pos {p : × } (hp_fst : 0 < p.1) :
(fun (x : × ) => x.1 ^ x.2) =ᶠ[nhds p] fun (x : × ) => Real.exp (Real.log x.1 * x.2)
theorem Real.continuousAt_rpow_of_ne (p : × ) (hp : p.1 0) :
ContinuousAt (fun (p : × ) => p.1 ^ p.2) p
theorem Real.continuousAt_rpow_of_pos (p : × ) (hp : 0 < p.2) :
ContinuousAt (fun (p : × ) => p.1 ^ p.2) p
theorem Real.continuousAt_rpow (p : × ) (h : p.1 0 0 < p.2) :
ContinuousAt (fun (p : × ) => p.1 ^ p.2) p
theorem Real.continuousAt_rpow_const (x : ) (q : ) (h : x 0 0 < q) :
ContinuousAt (fun (x : ) => x ^ q) x
theorem Filter.Tendsto.rpow {α : Type u_1} {l : Filter α} {f : α} {g : α} {x : } {y : } (hf : Filter.Tendsto f l (nhds x)) (hg : Filter.Tendsto g l (nhds y)) (h : x 0 0 < y) :
Filter.Tendsto (fun (t : α) => f t ^ g t) l (nhds (x ^ y))
theorem Filter.Tendsto.rpow_const {α : Type u_1} {l : Filter α} {f : α} {x : } {p : } (hf : Filter.Tendsto f l (nhds x)) (h : x 0 0 p) :
Filter.Tendsto (fun (a : α) => f a ^ p) l (nhds (x ^ p))
theorem ContinuousAt.rpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : f x 0 0 < g x) :
ContinuousAt (fun (t : α) => f t ^ g t) x
theorem ContinuousWithinAt.rpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {s : Set α} {x : α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : f x 0 0 < g x) :
ContinuousWithinAt (fun (t : α) => f t ^ g t) s x
theorem ContinuousOn.rpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h : xs, f x 0 0 < g x) :
ContinuousOn (fun (t : α) => f t ^ g t) s
theorem Continuous.rpow {α : Type u_1} [TopologicalSpace α] {f : α} {g : α} (hf : Continuous f) (hg : Continuous g) (h : ∀ (x : α), f x 0 0 < g x) :
Continuous fun (x : α) => f x ^ g x
theorem ContinuousWithinAt.rpow_const {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {x : α} {p : } (hf : ContinuousWithinAt f s x) (h : f x 0 0 p) :
ContinuousWithinAt (fun (x : α) => f x ^ p) s x
theorem ContinuousAt.rpow_const {α : Type u_1} [TopologicalSpace α] {f : α} {x : α} {p : } (hf : ContinuousAt f x) (h : f x 0 0 p) :
ContinuousAt (fun (x : α) => f x ^ p) x
theorem ContinuousOn.rpow_const {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {p : } (hf : ContinuousOn f s) (h : xs, f x 0 0 p) :
ContinuousOn (fun (x : α) => f x ^ p) s
theorem Continuous.rpow_const {α : Type u_1} [TopologicalSpace α] {f : α} {p : } (hf : Continuous f) (h : ∀ (x : α), f x 0 0 p) :
Continuous fun (x : α) => f x ^ p

Continuity results for cpow, part II #

These results involve relating real and complex powers, so cannot be done higher up.

theorem Complex.continuousAt_cpow_of_re_pos {p : × } (h₁ : 0 p.1.re p.1.im 0) (h₂ : 0 < p.2.re) :
ContinuousAt (fun (x : × ) => x.1 ^ x.2) p

See also continuousAt_cpow for a version that assumes p.1 ≠ 0 but makes no assumptions about p.2.

theorem Complex.continuousAt_cpow_const_of_re_pos {z : } {w : } (hz : 0 z.re z.im 0) (hw : 0 < w.re) :
ContinuousAt (fun (x : ) => x ^ w) z

See also continuousAt_cpow_const for a version that assumes z ≠ 0 but makes no assumptions about w.

theorem Complex.continuousAt_ofReal_cpow (x : ) (y : ) (h : 0 < y.re x 0) :
ContinuousAt (fun (p : × ) => p.1 ^ p.2) (x, y)

Continuity of (x, y) ↦ x ^ y as a function on ℝ × ℂ.

theorem Complex.continuousAt_ofReal_cpow_const (x : ) (y : ) (h : 0 < y.re x 0) :
ContinuousAt (fun (a : ) => a ^ y) x
theorem Complex.continuous_ofReal_cpow_const {y : } (hs : 0 < y.re) :
Continuous fun (x : ) => x ^ y

Limits and continuity for ℝ≥0 powers #

theorem NNReal.continuousAt_rpow {x : NNReal} {y : } (h : x 0 0 < y) :
ContinuousAt (fun (p : NNReal × ) => p.1 ^ p.2) (x, y)
theorem NNReal.eventually_pow_one_div_le (x : NNReal) {y : NNReal} (hy : 1 < y) :
∀ᶠ (n : ) in Filter.atTop, x ^ (1 / n) y
theorem Filter.Tendsto.nnrpow {α : Type u_1} {f : Filter α} {u : αNNReal} {v : α} {x : NNReal} {y : } (hx : Filter.Tendsto u f (nhds x)) (hy : Filter.Tendsto v f (nhds y)) (h : x 0 0 < y) :
Filter.Tendsto (fun (a : α) => u a ^ v a) f (nhds (x ^ y))
theorem NNReal.continuousAt_rpow_const {x : NNReal} {y : } (h : x 0 0 y) :
ContinuousAt (fun (z : NNReal) => z ^ y) x
theorem NNReal.continuous_rpow_const {y : } (h : 0 y) :
Continuous fun (x : NNReal) => x ^ y

Continuity for ℝ≥0∞ powers #

theorem ENNReal.eventually_pow_one_div_le {x : ENNReal} (hx : x ) {y : ENNReal} (hy : 1 < y) :
∀ᶠ (n : ) in Filter.atTop, x ^ (1 / n) y
theorem ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos {c : ENNReal} (hc : c ) {y : } (hy : 0 < y) :
Filter.Tendsto (fun (x : ENNReal) => c * x ^ y) (nhds 0) (nhds 0)
theorem Filter.Tendsto.ennrpow_const {α : Type u_1} {f : Filter α} {m : αENNReal} {a : ENNReal} (r : ) (hm : Filter.Tendsto m f (nhds a)) :
Filter.Tendsto (fun (x : α) => m x ^ r) f (nhds (a ^ r))