Understanding epsilon positive.

Question

I have a very basic question about $\epsilon$ which confuses me. Suppose we have somewhere $f \geq \epsilon ~ \forall ~ \epsilon >0$. Can we conclude that $f(x)>0$ or we can just say $f(x) \geq 0$. Similarly let $\epsilon >0$ be arbitrary, can we still say $\frac{1}{\epsilon}< \infty ~ \forall ~ \epsilon>0$ ?. If there are any references where the similar situation arises. It will be appreciated. Thank you.