mathoverflow_347178.variants.bounded_only

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Does the equality $$\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$$ hold when both suprema are finite?

theorem mathoverflow_347178.variants.bounded_only :
    answer(sorry) ↔ ∀ᵉ (n ≥ 2) (f : ℝ^n → ℝ) (hf : ContDiff ℝ 1 f)
        (h : BddAbove (range f)) (h' : BddAbove (range (fun x ↦ f (x + gradient f x)))),
        (⨆ x, f x) = ⨆ x, f (x + gradient f x)

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