Thread: 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

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

Since (I will assume that , but the other case is similar) let . Now choose , and the appropriate . Then for you have that .