deriv_pos_left_of_sign_deriv — Mathlib

English

If the left-hand side of b has derivative nonnegative, then f'(x) ≥ 0 for x<b near b.

Русский

Если слева от b производная неотрицательна, то для x<b ближе к b производная неотрицательна.

LaTeX

$$$$\\forall b\\text{ near } x_0\\ (x

Lean4

theorem deriv_pos_left_of_sign_deriv {f : ℝ → ℝ} {x₀ : ℝ}
    (h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x₀ - x)) : ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b > 0 :=
  by
  filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)
  rwa [← sub_pos, ← sign_eq_one_iff, ← hx', sign_eq_one_iff] at hx

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