File: Mathlib/Analysis/Calculus/DiffContOnCl.lean

full_name: DiffContOnCl
code: /-- A predicate saying that a function is differentiable on a set and is continuous on its
closure. This is a common assumption in complex analysis. -/
structure DiffContOnCl (f : E → F) (s : Set E) : Prop where
  protected differentiableOn : DifferentiableOn 𝕜 f s
  protected continuousOn : ContinuousOn f (closure s)
goal: Prop

full_name: DiffContOnCl.differentiableOn
code: /-- A predicate saying that a function is differentiable on a set and is continuous on its
closure. This is a common assumption in complex analysis. -/
structure DiffContOnCl (f : E → F) (s : Set E) : Prop where
  protected differentiableOn : DifferentiableOn 𝕜 f s
  protected continuousOn : ContinuousOn f (closure s)
goal: DifferentiableOn 𝕜 f s

full_name: DiffContOnCl.continuousOn
code: /-- A predicate saying that a function is differentiable on a set and is continuous on its
closure. This is a common assumption in complex analysis. -/
structure DiffContOnCl (f : E → F) (s : Set E) : Prop where
  protected differentiableOn : DifferentiableOn 𝕜 f s
  protected continuousOn : ContinuousOn f (closure s)
goal: ContinuousOn f (closure s)

full_name: DifferentiableOn.diffContOnCl
code: theorem DifferentiableOn.diffContOnCl (h : DifferentiableOn 𝕜 f (closure s)) : DiffContOnCl 𝕜 f s 
goal: DiffContOnCl 𝕜 f s

full_name: Differentiable.diffContOnCl
code: theorem Differentiable.diffContOnCl (h : Differentiable 𝕜 f) : DiffContOnCl 𝕜 f s 
goal: DiffContOnCl 𝕜 f s

full_name: IsClosed.diffContOnCl_iff
code: theorem IsClosed.diffContOnCl_iff (hs : IsClosed s) : DiffContOnCl 𝕜 f s ↔ DifferentiableOn 𝕜 f s 
goal: DiffContOnCl 𝕜 f s ↔ DifferentiableOn 𝕜 f s

full_name: diffContOnCl_univ
code: theorem diffContOnCl_univ : DiffContOnCl 𝕜 f univ ↔ Differentiable 𝕜 f 
goal: DiffContOnCl 𝕜 f Set.univ ↔ Differentiable 𝕜 f

full_name: diffContOnCl_const
code: theorem diffContOnCl_const {c : F} : DiffContOnCl 𝕜 (fun _ : E => c) s 
goal: DiffContOnCl 𝕜 (fun x => c) s

full_name: DiffContOnCl.comp
code: theorem comp {g : G → E} {t : Set G} (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g t)
    (h : MapsTo g t s) : DiffContOnCl 𝕜 (f ∘ g) t 
goal: DiffContOnCl 𝕜 (f ∘ g) t

full_name: DiffContOnCl.continuousOn_ball
code: theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
    ContinuousOn f (closedBall x r) 
goal: ContinuousOn f (Metric.closedBall x r)

full_name: DiffContOnCl.mk_ball
code: theorem mk_ball {x : E} {r : ℝ} (hd : DifferentiableOn 𝕜 f (ball x r))
    (hc : ContinuousOn f (closedBall x r)) : DiffContOnCl 𝕜 f (ball x r) 
goal: DiffContOnCl 𝕜 f (Metric.ball x r)

full_name: DiffContOnCl.differentiableAt
code: protected theorem differentiableAt (h : DiffContOnCl 𝕜 f s) (hs : IsOpen s) (hx : x ∈ s) :
    DifferentiableAt 𝕜 f x 
goal: DifferentiableAt 𝕜 f x

full_name: DiffContOnCl.differentiableAt'
code: theorem differentiableAt' (h : DiffContOnCl 𝕜 f s) (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x 
goal: DifferentiableAt 𝕜 f x

full_name: DiffContOnCl.mono
code: protected theorem mono (h : DiffContOnCl 𝕜 f s) (ht : t ⊆ s) : DiffContOnCl 𝕜 f t 
goal: DiffContOnCl 𝕜 f t

full_name: DiffContOnCl.add
code: theorem add (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s) : DiffContOnCl 𝕜 (f + g) s 
goal: DiffContOnCl 𝕜 (f + g) s

full_name: DiffContOnCl.add_const
code: theorem add_const (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => f x + c) s 
goal: DiffContOnCl 𝕜 (fun x => f x + c) s

full_name: DiffContOnCl.const_add
code: theorem const_add (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => c + f x) s 
goal: DiffContOnCl 𝕜 (fun x => c + f x) s

full_name: DiffContOnCl.neg
code: theorem neg (hf : DiffContOnCl 𝕜 f s) : DiffContOnCl 𝕜 (-f) s 
goal: DiffContOnCl 𝕜 (-f) s

full_name: DiffContOnCl.sub
code: theorem sub (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s) : DiffContOnCl 𝕜 (f - g) s 
goal: DiffContOnCl 𝕜 (f - g) s

full_name: DiffContOnCl.sub_const
code: theorem sub_const (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => f x - c) s 
goal: DiffContOnCl 𝕜 (fun x => f x - c) s

full_name: DiffContOnCl.const_sub
code: theorem const_sub (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => c - f x) s 
goal: DiffContOnCl 𝕜 (fun x => c - f x) s

full_name: DiffContOnCl.const_smul
code: theorem const_smul {R : Type*} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F]
    [ContinuousConstSMul R F] (hf : DiffContOnCl 𝕜 f s) (c : R) : DiffContOnCl 𝕜 (c • f) s 
goal: DiffContOnCl 𝕜 (c • f) s

full_name: DiffContOnCl.smul
code: theorem smul {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F]
    [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {f : E → F} {s : Set E} (hc : DiffContOnCl 𝕜 c s)
    (hf : DiffContOnCl 𝕜 f s) : DiffContOnCl 𝕜 (fun x => c x • f x) s 
goal: DiffContOnCl 𝕜 (fun x => c x • f x) s

full_name: DiffContOnCl.smul_const
code: theorem smul_const {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜']
    [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {s : Set E} (hc : DiffContOnCl 𝕜 c s)
    (y : F) : DiffContOnCl 𝕜 (fun x => c x • y) s 
goal: DiffContOnCl 𝕜 (fun x => c x • y) s

full_name: DiffContOnCl.inv
code: theorem inv {f : E → 𝕜} (hf : DiffContOnCl 𝕜 f s) (h₀ : ∀ x ∈ closure s, f x ≠ 0) :
    DiffContOnCl 𝕜 f⁻¹ s 
goal: DiffContOnCl 𝕜 f⁻¹ s

full_name: Differentiable.comp_diffContOnCl
code: theorem Differentiable.comp_diffContOnCl {g : G → E} {t : Set G} (hf : Differentiable 𝕜 f)
    (hg : DiffContOnCl 𝕜 g t) : DiffContOnCl 𝕜 (f ∘ g) t 
goal: DiffContOnCl 𝕜 (f ∘ g) t

full_name: DifferentiableOn.diffContOnCl_ball
code: theorem DifferentiableOn.diffContOnCl_ball {U : Set E} {c : E} {R : ℝ} (hf : DifferentiableOn 𝕜 f U)
    (hc : closedBall c R ⊆ U) : DiffContOnCl 𝕜 f (ball c R) 
goal: DiffContOnCl 𝕜 f (Metric.ball c R)