Let $f: [a,b]^n \mapsto [a,b]$ be a L-Lipschitz continuous function (w.r.t. to the distance function induced by some $l_p$-norm) over a bounded domain where $a, b \in \mathbb{R}$.
If $g = \frac{\partial f}{\partial x_i}$ is a partial derivate of $f$ will $g$ be Lipschitz continuous? and if so can we relate its Lipschitz constant to that of of $f$?
No. Let $n=1$, $[a,b]=[0,1]$ and $f(x)=x^{3/2}$.
We have $f'(x)=\frac{3}{2} \sqrt{x}$
By the mean value theorem:
$|f(x)-f(y)| \le \frac{3}{2}|x-y|$.
But $f'$ is not Lipschitz cont.
To this end assume that there is $L \ge 0$ such that $|f'(x)-f'(y)| \le |x-y|$
If $y=0$ this gives $\sqrt{x} \le Lx$ for all $x \in [0,1]$.
For $x \in (0,1]$ we derive the contradiction
$1 \le L \sqrt{x}$