Let $x$, $y$ be numbers in $(0, \delta)$.
$ \left|\left(|y|^{1/2} - |x|^{1/2}\right)\left(|y|^{1/2} + |x|^{1/2}\right)\right| = \left||y| - |x|\right| = |y - x| $
Therefore: $ |f(y) - f(x)| = \frac{|y - x|}{\left||y|^{1/2} + |x|^{1/2}\right|} \tag{1} $
If $f(x)=|x|^{1/2}$ is Lipschitz continuous, we can find $K > 0$ so that:
$ |f(y) - f(x)| \le K|y - x| \tag{2} $
Put (1) and (2) together to get:
$ \frac{1}{K} \le \left||y|^{1/2} + |x|^{1/2}\right| $
By making $x$ and $y$ approach $0$, we can make the RHS as small as we desire. Thus, we have a contradiction.