Stronger Whitney approximation for smooth manifolds

Question

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?

Answer 1

As it was pointed out to me, this (almost exact) statement is proven in Differential Manifolds by Kosinski (Chapter III, Theorem 2.5).