36 lines
798 B
Lean4
36 lines
798 B
Lean4
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
|