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Section 9.6: fix Exercise 9.6.1 statements and add Exercise 9.6.2 #488

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teorth merged 1 commit into from
Apr 21, 2026
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Section 9.6: fix Exercise 9.6.1 statements and add Exercise 9.6.2 #488

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32 changes: 23 additions & 9 deletions Analysis/Section_9_6.lean
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Expand Up @@ -154,26 +154,40 @@ theorem sInf.of_continuous_on_compact {a b:ℝ} (h:a < b) (f:ℝ → ℝ) (hf: C
choose x hx h' using IsMinOn.of_continuous_on_compact h hf
grind [sInf.of_isMinOn]

/-- Exercise 9.6.1 -/
/-- Exercise 9.6.1 a) -/
example : ∃ f: ℝ → ℝ, ContinuousOn f (.Ioo 1 2) ∧ BddOn f (.Ioo 1 2) ∧
∃ x₀ ∈ Set.Ioo 1 2, IsMinOn f (.Ioo 1 2) x₀ ∧
¬ ∃ x₀ ∈ Set.Ioo 1 2, IsMaxOn f (.Ioo 1 2) x₀
:= by sorry

/-- Exercise 9.6.1 -/
example : ∃ f: ℝ → ℝ, ContinuousOn f (.Ioo 1 2) ∧ BddOn f (.Ioo 1 2) ∧
∃ x₀ ∈ Set.Ioo 1 2, IsMaxOn f (.Ioo 1 2) x₀ ∧
¬ ∃ x₀ ∈ Set.Ioo 1 2, IsMinOn f (.Ioo 1 2) x₀
/-- Exercise 9.6.1 b) -/
example : ∃ f: ℝ → ℝ, ContinuousOn f (.Ici 0) ∧ BddOn f (.Ici 0) ∧
∃ x₀ ∈ Set.Ici 0, IsMaxOn f (.Ici 0) x₀ ∧
¬ ∃ x₀ ∈ Set.Ici 0, IsMinOn f (.Ici 0) x₀
:= by sorry

/-- Exercise 9.6.1 -/
/-- Exercise 9.6.1 c) -/
example : ∃ f: ℝ → ℝ, BddOn f (.Icc (-1) 1) ∧
¬ ∃ x₀ ∈ Set.Icc (-1) 1, IsMinOn f (.Icc (-1) 1) x₀ ∧
¬ ∃ x₀ ∈ Set.Icc (-1) 1, IsMaxOn f (.Icc (-1) 1) x₀
(¬ ∃ x₀ ∈ Set.Icc (-1) 1, IsMinOn f (.Icc (-1) 1) x₀) ∧
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@rkirov rkirov Apr 19, 2026

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this one was subtle, without parens lean reads it as ¬ (A ∧ ¬ B)

(¬ ∃ x₀ ∈ Set.Icc (-1) 1, IsMaxOn f (.Icc (-1) 1) x₀)
:= by sorry

/-- Exercise 9.6.1 -/
/-- Exercise 9.6.1 d) -/
example : ∃ f: ℝ → ℝ, ¬ BddAboveOn f (.Icc (-1) 1) ∧ ¬ BddBelowOn f (.Icc (-1) 1) := by sorry

/-- Exercise 9.6.2 -/
theorem BddOn.add (f g : ℝ → ℝ) (X : Set ℝ) (hf : BddOn f X) (hg : BddOn g X) :
BddOn (f + g) X := by sorry

theorem BddOn.sub (f g : ℝ → ℝ) (X : Set ℝ) (hf : BddOn f X) (hg : BddOn g X) :
BddOn (f - g) X := by sorry

theorem BddOn.mul (f g : ℝ → ℝ) (X : Set ℝ) (hf : BddOn f X) (hg : BddOn g X) :
BddOn (f * g) X := by sorry

def BddOn.div : Decidable (∀ (f g : ℝ → ℝ) (X : Set ℝ) (_ : ∀ x ∈ X, g x ≠ 0) (_ : BddOn f X)
(_: BddOn g X), (BddOn (f / g) X)) := by
-- the first line of this construction should be either `apply isTrue` or `apply isFalse`, depending on whether you believe the given statement to be true or false.
sorry

end Chapter9
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