Boundedness of gradient Let $f:\mathbb{R}^N\to\mathbb{R}^N$ is such that $|\langle f(x),y\rangle|\leq C|x||y|$ holds for all $x,y\in\mathbb{R}^N$.
Then, there exists a fixed positive constant $C$ such that
$$
|f(x)|\leq C|x|,\,\forall x\in\mathbb{R}^N.
$$
I tried to prove this way: We can write
$$
|f(x)|=\sup_{y\in\mathbb{R}^N\setminus\{0\}}\frac{|\langle f(x),y\rangle|}{|y|}.
$$
Hence, using the given condition $|\langle f(x),y\rangle|\leq C|x||y|$, we get $|f(x)|\leq\frac{C|x||y|}{|y|}=C|x|$. Therefore, the result follows.
Please inform me if I am missing something. Thanks.
Since The hypothesis is true for all $\mathbf x, \mathbf y \in \mathbb R^n$ then it's true for $\mathbf y = \mathbf 0$.
So let $\mathbf y = \mathbf 0$.
$$\Vert ( \mathbf f( \mathbf x),\mathbf y)\Vert= \Vert ( \mathbf f( \mathbf x),\mathbf 0)\Vert = \Vert\mathbf f( \mathbf x)\Vert \le C \,\Vert \mathbf x\Vert \, \Vert\mathbf y\Vert= C \,\Vert \mathbf x\Vert \, \Vert\mathbf 0\Vert=0.$$
Hence $\Vert\mathbf f( \mathbf x)\Vert \le 0$, which implies that $\Vert\mathbf f( \mathbf x)\Vert = 0$.
Therefore, for all $\mathbf x \in \mathbb R^n$, $\Vert\mathbf f( \mathbf x)\Vert \le C$ for any $C \ge 0$.

Edit:
I don't know whether it was supposed to be the case that $\mathbf y \neq \mathbf 0$; in that case take $\mathbf y = \mathbf e_i$, so:
$$\Vert \mathbf f(\mathbf x)\Vert \le \Vert (\mathbf f (\mathbf x),\mathbf e_i)\Vert \le C \, \Vert \mathbf x\Vert \, \Vert \mathbf e_i\Vert=C \, \Vert \mathbf x\Vert$$
Since The hypothesis is true for all $\mathbf x, \mathbf y \in \mathbb R^n$ then it's true for $\mathbf y = \mathbf 0$.
So let $\mathbf y = \mathbf 0$.
$$\Vert ( \mathbf f( \mathbf x),\mathbf y)\Vert= \Vert ( \mathbf f( \mathbf x),\mathbf 0)\Vert = \Vert\mathbf f( \mathbf x)\Vert \le C \,\Vert \mathbf x\Vert \, \Vert\mathbf y\Vert= C \,\Vert \mathbf x\Vert \, \Vert\mathbf 0\Vert=0.$$
Hence $\Vert\mathbf f( \mathbf x)\Vert \le 0$, which implies that $\Vert\mathbf f( \mathbf x)\Vert = 0$.
Therefore, for all $\mathbf x \in \mathbb R^n$, $\Vert\mathbf f( \mathbf x)\Vert \le C$ for any $C \ge 0$.

Edit:
I don't know whether it was supposed to be the case that $\mathbf y \neq \mathbf 0$; in that case take $\mathbf y = \mathbf e_i$, so:
$$\Vert \mathbf f(\mathbf x)\Vert \le \Vert (\mathbf f (\mathbf x),\mathbf e_i)\Vert \le C \, \Vert \mathbf x\Vert \, \Vert \mathbf e_i\Vert=C \, \Vert \mathbf x\Vert$$