intermediate_value_Icc — Mathlib

English

Equivalent expression: IsTotallyDisconnected s ⇔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y.

Русский

Эквивалентно: IsTotallyDisconnected s ⇔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y.

LaTeX

$$IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y$$

Lean4

/-- **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≤ t ≤ f b`. -/
theorem intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
    Icc (f a) (f b) ⊆ f '' Icc a b :=
  isPreconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf

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