R=Reals
Let f : R --> R be continuous on R, and let P := {x in R : f(x) is greater than 0}. If c is in P, show that there exists a neighborhood V delta(c) as an element of P.
Let $\displaystyle p$ such that $\displaystyle r=f(p)>0$ and $\displaystyle f$ is continous at $\displaystyle p$ then For $\displaystyle \epsilon =r$ there exist a $\displaystyle \delta >0$ such that $\displaystyle \vert f(p)-f(x) \vert < r$ whenever $\displaystyle \vert x-p \vert < \delta$. And so with this $\displaystyle \delta$ we have $\displaystyle f(x)>0$