What don't you understand about the statement? It's saying suppose that you found some point such that yet can you find some neighborhood of such that and are neither zero anywhere on that neighborhood except is of course zero at . Anyways..

Since and is continuous there exists an open ball such that does not vanish on that open ball. Now, if vanished somewhere else on that open ball besides you could apply Rolle's theorem to find a zero of on the open ball contradictory to construction.