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**paulrb** Thanks...although we have not covered what an open ball is in the class. We have covered Rolle's theorem though.

Intuitively I think I understand now. Since $\displaystyle g'(x_0) \neq 0$, there has to be some point near $\displaystyle x_0$where g is not 0. Since g is continuous, all of the points between $\displaystyle x_0$ and this point are nonzero. Furthermore for g to reach this nonzero point from 0, there has to be an interval between them where g'(x) is nonzero, since g'(x) is continuous. Although I still don't know how to prove this rigorously.