mathoverflow_347178.variants.bounded_iff

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Is the boundedness of $\sup_{x \in \mathbb R^n}f(x)$ and $\sup_{x\in \mathbb R^n} f(x+\nabla f(x))$ equivalent?

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

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