In Murdock's Pertubations problem 1.7.4, we're looking at the equation $$\phi(x,\tau,\epsilon) = (x-1)(x-\tau)+\epsilon x = 0 $$ rescaled by $x=y\epsilon^\nu+1$ and $\sigma = \tau\epsilon^\lambda+1$ for $\nu=0.5$ and $\lambda=1$ to get: $$y^2-\sqrt\epsilon(\sigma+1)y-1=0$$
Now in the reduced problem where $\epsilon=0$ we have $y^2-1=0$ so $y=\pm1$ independent of $\sigma$ and hence also of $\tau$.
What are the regions of uniformity? Since y doesn't depend on $\sigma$ I can't apply Theorem 1.7.1 since it fails condition 1.7.6 (I think).
What else am I supposed to use? Is there just no region of uniformity?