bounded away from zero vs positive

Question

As a follow on to this question, are the following equivalent:

1. $\exists c > 0 : f(x) \geq c$
2. $f(x) > 0$

for some function $f: X\to\mathbb{R}$.

How about if $f \in \mathcal{C}^1$? Further, what conditions would lead to equivalence?

Answer 1

No: it can happen that $f(x)>0$ for all $x \in X$, but $\inf_{x \in X} f(x)=0$. For a compact set $X$ this would require that $f$ be discontinuous, but for a non-compact set this isn't necessary. For an example, consider $f(x)=1/x$,$X=[1,\infty)$.