# Math Help - real analysis hw question

1. ## real analysis hw question

i need to prove: if f is continuous on EcR and if f(x) is not equal to 0 for some point x in E then there is a nbd Q of x such that f(y) is not equal to 0 for all y in the intersection of Q and E

any help? thanks so much

2. Unless there is something that I am missing this can be derived directly from the definition of continuity as follows.

Since $f(x_0)\neq 0$ (I will assume that $f(x_0)>0$, but the other case is similar) let $f(x_0)=\alpha$. Now choose $\epsilon=\frac{\alpha}{2}$, and the appropriate $\delta$. Then for $|x-x_0|<\delta$ you have that $0.