diff --git a/Analysis/Section_9_4.lean b/Analysis/Section_9_4.lean index 4a35f265..21366eb7 100644 --- a/Analysis/Section_9_4.lean +++ b/Analysis/Section_9_4.lean @@ -29,7 +29,7 @@ theorem ContinuousWithinAt.iff (X:Set ℝ) (f: ℝ → ℝ) (x₀:ℝ) : #check continuousOn_univ #check continuousWithinAt_univ -/-- Example 9.4.2 --/ +/-- Example 9.4.2. -/ example (c x₀:ℝ) : ContinuousWithinAt (fun x ↦ c) .univ x₀ := by sorry example (c x₀:ℝ) : ContinuousAt (fun x ↦ c) x₀ := by sorry @@ -38,18 +38,18 @@ example (c:ℝ) : ContinuousOn (fun x:ℝ ↦ c) .univ := by sorry example (c:ℝ) : Continuous (fun x:ℝ ↦ c) := by sorry -/-- Example 9.4.3 --/ +/-- Example 9.4.3. -/ example : Continuous (fun x:ℝ ↦ x) := by sorry -/-- Example 9.4.4 --/ +/-- Example 9.4.4. -/ example {x₀:ℝ} (h: x₀ ≠ 0) : ContinuousAt Real.sign x₀ := by sorry example :¬ ContinuousAt Real.sign 0 := by sorry -/-- Example 9.4.5 --/ +/-- Example 9.4.5. -/ example (x₀:ℝ) : ¬ ContinuousAt f_9_3_21 x₀ := by sorry -/-- Example 9.4.6 --/ +/-- Example 9.4.6. -/ noncomputable abbrev f_9_4_6 (x:ℝ) : ℝ := if x ≥ 0 then 1 else 0 example {x₀:ℝ} (h: x₀ ≠ 0) : ContinuousAt f_9_4_6 x₀ := by sorry @@ -68,7 +68,7 @@ theorem ContinuousWithinAt.tfae (X:Set ℝ) (f: ℝ → ℝ) (x₀:ℝ) : ].TFAE := by sorry -/-- Remark 9.4.8 --/ +/-- Remark 9.4.8. -/ theorem _root_.Filter.Tendsto.comp_of_continuous {X:Set ℝ} {f: ℝ → ℝ} {x₀:ℝ} (h_cont: ContinuousWithinAt f X x₀) {a: ℕ → ℝ} (ha: ∀ n, a n ∈ X) (hconv: Filter.atTop.Tendsto a (nhds x₀)):