This proof is left as an exercise in the book:

Suppose that g is continuously differentiable in an interval about $\displaystyle x_0$ and $\displaystyle g(x_0) = 0$. If $\displaystyle g'(x_0) \neq 0$, show that there is a small interval about $\displaystyle x_0$ in which neither g nor g' vanish except for the root of g at the point $\displaystyle x_0$.

I don't really know what this is saying, nor how to prove to prove it.