Initial checkout

ColloquiumLean/ContinuousSquareSolution.lean

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+-- Library Import: Basic facts about real numbers and the root function
+import Mathlib.Data.Real.Sqrt
+open Real
+
+-- In the following, x and y are real numbers
+variable {x : ℝ} {y : ℝ}
+
+-- Definition: Function (f : ℝ → ℝ) is continuous at (x₀ : ℝ)
+def continuous (f : ℝ → ℝ) (x₀ : ℝ) :=
+  ∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| < δ → |f x - f x₀| < ε
+
+-- Definition: The square function
+def squareFct : ℝ → ℝ := fun x ↦ x ^ 2
+
+
+-- Reminder: Three facts from the library that we will use.
+example : 0 < x → 0 < √x := sqrt_pos.2
+
+example : x ^ 2 < y  ↔  (-√y < x) ∧ (x < √y) := sq_lt
+
+example : |x| < y  ↔  -y < x ∧ x < y := abs_lt
+
+
+-- Theorem: The square function is continuous at x₀ = 0
+theorem ContinuousAt_sq :
+    continuous squareFct 0 := by
+
+  unfold continuous
+
+  intro ε hε
+  use √ε
+
+  constructor
+  · apply sqrt_pos.2
+    exact hε
+  · intro x hx
+    simp_all [squareFct]
+    apply sq_lt.2
+    exact abs_lt.1 hx