Definition:(Hale,1980) A function $f(x)$ defined in a domain $D$ in $R^{n}$ is said to be locally Lipschitzian in $x$ if for any closed bounded set $U$ in $D$ there is a $k=k_{U}$ such that $|f(x)-f(y)|\leq k|x-y|$ for $x, y$ in $U$.
(I need work with this definition.)
With above definition, is the function $f(x,y)=xy^c$, $0\leq x,y$ and $c=\frac{a}{b},$ with $a,b$ positive integers, locally Lipschitzian ?
Thanks.