I was reading John Lee's Introduction to Smooth manifolds, and I came across this question:
- Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous function. Using a partition of unity, show that there is a smooth function $\tilde{\delta} : M \rightarrow \mathbb{R}$ such that $0 < \tilde{\delta}(x) < \delta(x)$ for all $x \in M$.
I thought about it for a while, and I'm pretty stuck on it. Does anyone have any ideas?
(Edit: It has been pointed out that one can basically assume that $M = \mathbb{R}$, because the proof should be the same in both cases. If you do not know anything about smooth manifolds, feel free to do this!)