theorem extracted_formal_statement_157 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} {b : Fin ↑m → ℝ} {μ : ℝ} {μpos : 0 < μ}
  {Ane0 : A ≠ 0} {x₀ : EuclideanSpace ℝ (Fin ↑n)} (p : LASSO A b μ μpos Ane0 x₀) (x : EuclideanSpace ℝ (Fin ↑n))
  (a : x ∈ Set.univ) (h_original_goal : ContinuousWithinAt p.h Set.univ x) (ε : ℝ) :
  ε > 0 → ∃ δ > 0, ∀ (x' : EuclideanSpace ℝ (Fin ↑n)), ‖x - x'‖ ≤ δ → ‖p.h x' - p.h x‖ ≤ ε := sorry

State before:
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
b : Fin ↑m → ℝ
μ : ℝ
μpos : 0 < μ
Ane0 : A ≠ 0
x₀ : EuclideanSpace ℝ (Fin ↑n)
p : LASSO A b μ μpos Ane0 x₀
x : EuclideanSpace ℝ (Fin ↑n)
a✝ : x ∈ Set.univ
⊢ ContinuousWithinAt p.h Set.univ x

Tactic: rw [continuousWithinAt_univ, ContinuousAt_iff_Convergence]

State after:
[n m : ℕ+
 A : Matrix (Fin ↑m) (Fin ↑n) ℝ
 b : Fin ↑m → ℝ
 μ : ℝ
 μpos : 0 < μ
 Ane0 : A ≠ 0
 x₀ : EuclideanSpace ℝ (Fin ↑n)
 p : LASSO A b μ μpos Ane0 x₀
 x : EuclideanSpace ℝ (Fin ↑n)
 a✝ : x ∈ Set.univ
 ⊢ ∀ ε > 0, ∃ δ > 0, ∀ (x' : EuclideanSpace ℝ (Fin ↑n)), ‖x - x'‖ ≤ δ → ‖p.h x' - p.h x‖ ≤ ε]

Relevant lemmas:
continuousWithinAt_univ : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β]
  (f : α → β) (x : α), ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x
ContinuousAt_iff_Convergence : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : E → ℝ} {x : E},
  ContinuousAt f x ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x - x'‖ ≤ δ → ‖f x' - f x‖ ≤ ε