Question: Let $f: \mathbb{R}^{d} \to \mathbb{R}$ be a smooth strictly convex function with the growth condition $|f(x)|/|x| \to \infty \ (|x| \to \infty)$. Is the gradient $\nabla f$ surjective?
My attempt:I could observe that $\nabla f(x)$ diverges as $x$ goes to $\infty$ along with any direction as follows.
By the convexity, it holds$f(x) - f(y) \leq \nabla f(x)\cdot (x-y)$.By the growth condition, it follows $\nabla f(x)\cdot(x-y)/|x| \geq (f(x)-f(y))/|x| \to \infty \ (|x| \to \infty)$.Hence, we have $\nabla f(x)\cdot x/|x| \to \infty \ (|x| \to \infty)$.This especially implies $\nabla f(ce_{i}) \to \pm\infty \ (c \to \pm\infty)$, where $e_{i}:=(\delta_{ij})_{1 \leq i \leq d}$.When $d=1$, this solves my question by the intermediate value theorem, but I don't know what to do in the multi-dimensional case.
Thank you for any comments.