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

Printable View

- Apr 22nd 2009, 12:42 PMnellygreal 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 - Apr 23rd 2009, 10:37 AMputnam120
Unless there is something that I am missing this can be derived directly from the definition of continuity as follows.

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