Stronger Whitney approximation for smooth manifolds

If $M$ and $N$ are smooth manifolds, then any continuous map $f:M\rightarrow N$ is homotopic to a smooth map $g:M\rightarrow N$. If $f$ is smooth on a closed set $K\subseteq M$, the homotopy can be taken relative to $K$. (This is in proven in e.g. Lee).
Now assume that we have a metric $\rho$ on $N$. I was wondering if, additionally, $g$ can be taken to be a $\delta$-approximation of $f$ where $\delta:M\rightarrow (0,\infty)$ is a continuous map (i.e. $\rho(f(x),g(x))\leq \delta(x)$ for all $x\in M$).
There is a theorem in Metric Structures in Differential Geometry by Walschap (Theorem 1.2) that states something similar, but $N$ is assumed to be compact. Is this necessary?

• If you have acess to Kosinski's differnetial manifolds book, then theorem 2.5 (page 8) is close to what you want. The only differences are that his $\delta$ is constant, and the codomain is $\mathbb{R}^n$. – Jason DeVito Mar 1 '16 at 15:57
• Even better: page 47, again Theorem 2.5 (b)is exactly what you want. – Jason DeVito Mar 1 '16 at 16:15
• You are right, thanks a lot! – Pete Mar 1 '16 at 16:36
• You might want to post an answer to the question, to get the question off the unanswered list. – Jason DeVito Mar 1 '16 at 18:47