Let $M$ be a smooth manifold. Let $f : M \to \mathbb{R}$ be a positive continuous function. Then is it true that there exists a positive smooth function $g : M \to \mathbb{R}$ such that $0 < g \le f$?
This seems obvious, but I do not know how to approach. Thank you.
(Edit) I think I proved it. If $M$ is compact, then $f$ has minimum, say $m > 0$, so we can consider the constant function $g = m$. If $M$ is noncompact, then consider some nice partition $\{ K_i \}_i$ of $M$ by compact sets, and let $m_i > 0$ be the minimum of $f|_{K_i}$. Now we may take some nice smooth function $g$ such that $g \le m_i$ on each $K_i$.