Let $\mu > 0$, consider the Huber function $$H_{\mu}(x) = \left\{\begin{array}{lll} \dfrac{\|x\|^2}{2\mu}, &\quad& \|x\| \le \mu, \\ \|x\| - \dfrac{\mu}{2}, &\quad& \|x\| > \mu. \end{array}\right. $$ Show that $H_{\mu} \in C^{1,1}_{1/\mu}$.
Here, I have to prove $H_{\mu}$ is continuously differentiable function and derivative of $H_{\mu}$ is locally Lipschitz.
I know that $H_{\mu}$ is continuously differentiable on $\|x\| < \mu$ and on $\|x\| > \mu$. But on $B[0,\mu]$, I don't have any ideas to check, as well as proving that derivative of $H_{\mu}$ is locally Lipschitz.
Any help would be appreciated.