I'm trying to prove the following proposition.
Let M be a smooth manifold and let $g:M→R^+$ be a continuous function. Show that there is a smooth function $f:M→ R^+$ such that $f(x) < g(x)$ for all $x ∈ M$. (Hint: Use partition of unity on an open cover of M by coordinate balls.)
I've only come up with $f(x) := g(x)/2 $.However, considering the given hint, I'm guessing that does not work, and I don't know why.
As it is pointed out, since $g$ is not smooth, $f$ does not have to be as well, so the question is now, how to find $f$.