continuousOn_ball — Mathlib

English

A derivative sign hypothesis implying a local maximum at the point.

Русский

Гипотеза знака производной приводит к локальному максимуму в точке.

LaTeX

$$$$\\text{If the derivative sign condition holds, then the point is a local maximum.}$$$$

Lean4

theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
    ContinuousOn f (closedBall x r) := by
  rcases eq_or_ne r 0 with (rfl | hr)
  · rw [closedBall_zero]
    exact continuousOn_singleton f x
  · rw [← closure_ball x hr]
    exact h.continuousOn

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