Prove function do not satisfy Lipschitz condition

Question

How to show that for function $f:\mathbb{R}^n\mapsto\mathbb{R}$ $f(\mathbf x)=||\mathbf x||^{3/2}$,

the Lipschitz condition $||\nabla f(x)-\nabla f(y)||\le L||x-y||$ for all $x,\, y$ is not satisfied for any $L$?

Answer 1

First note that \begin{align} \nabla f(\mathbf{x}) = \frac{3}{2}\frac{\mathbf{x}}{\sqrt{\left\|\mathbf{x} \right\|}} \end{align} Take $\mathbf{y} = - \mathbf{x}$. Then \begin{align} \left\|\nabla f(\mathbf{x}) - \nabla f(\mathbf{y}) \right\| = 3\sqrt{\left\| \mathbf{x}\right\|} \end{align} Suppose there was an $L$ such that the inequality held. Then \begin{align} 3\sqrt{\left\| \mathbf{x}\right\|} \le 2L \left\|\mathbf{x}\right\| \end{align} so that \begin{align} L \ge \frac{2}{3\sqrt{\left\|\mathbf{x}\right\|}}. \end{align} But as $\left\|\mathbf{x}\right\| \to 0$ . . . whoops.