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Fix Exercise 9.8.2 IVT counterexample statements #495

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teorth merged 1 commit into from
Apr 30, 2026
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Fix Exercise 9.8.2 IVT counterexample statements #495

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26 changes: 19 additions & 7 deletions Analysis/Section_9_8.lean
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Expand Up @@ -107,13 +107,25 @@ theorem BddOn.of_antitone {a b:ℝ} {f:ℝ → ℝ} (hf: AntitoneOn f (.Icc a b)


/-- Exercise 9.8.2 -/
theorem no_strictmono_intermediate_value : ∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: StrictMonoOn f (.Icc a b)), ¬ ∃ y, y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f a) (f b) := by sorry

theorem no_monotone_intermediate_value : ∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: MonotoneOn f (.Icc a b)), ¬ ∃ y, y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f a) (f b) := by sorry

theorem no_strictanti_intermediate_value : ∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: StrictAntiOn f (.Icc a b)), ¬ ∃ y, y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f a) (f b) := by sorry

theorem no_antitone_intermediate_value : ∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: AntitoneOn f (.Icc a b)), ¬ ∃ y, y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f a) (f b) := by sorry
theorem no_strictmono_intermediate_value :
∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: StrictMonoOn f (.Icc a b)),
∃ y, (y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f b) (f a)) ∧
¬ ∃ c ∈ Set.Icc a b, f c = y := by sorry

theorem no_monotone_intermediate_value :
∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: MonotoneOn f (.Icc a b)),
∃ y, (y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f b) (f a)) ∧
¬ ∃ c ∈ Set.Icc a b, f c = y := by sorry

theorem no_strictanti_intermediate_value :
∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: StrictAntiOn f (.Icc a b)),
∃ y, (y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f b) (f a)) ∧
¬ ∃ c ∈ Set.Icc a b, f c = y := by sorry

theorem no_antitone_intermediate_value :
∃ (a b:ℝ) (hab: a < b) (f:ℝ → ℝ) (hf: AntitoneOn f (.Icc a b)),
∃ y, (y ∈ Set.Icc (f a) (f b) ∨ y ∈ Set.Icc (f b) (f a)) ∧
¬ ∃ c ∈ Set.Icc a b, f c = y := by sorry

/-- Exercise 9.8.3 -/
theorem mono_of_continuous_inj {a b:ℝ} (h: a < b) {f:ℝ → ℝ}
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