Initial checkout

ColloquiumLean/ContinuousSquare.lean

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+import Mathlib
+
+open Real
+
+variable
+  {x : ℝ} {y : ℝ}
+
+-- Definition: Function (f : ℝ → ℝ) is continuous at (x₀ : ℝ)
+def continuous_at (f : ℝ → ℝ) (x₀ : ℝ) :=
+  ∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| < δ → |f x - f x₀| < ε
+
+-- Three lemmas from the lean mathematical library
+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
+
+-- The square function
+def squareFct : ℝ → ℝ := fun x ↦ x ^ 2
+
+
+theorem continuous_at_squareFct :
+    continuous_at squareFct 0 := by
+  unfold continuous_at
+  intro e he
+  use √e
+  constructor
+  · apply sqrt_pos.2
+    exact he
+  · intro x hx
+    simp_all [squareFct]
+    apply sq_lt.2
+    apply abs_lt.1
+    exact hx