Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold *[with boundary*][1].  Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the topology of [uniform convergence on compacta][2].  



Intuitively it seems that every continuous function $f:M\rightarrow N$ can be approximated by continuous functions of the form $g:M\rightarrow\operatorname{int}(N)$; where $\operatorname{int}(N)=N- \partial N$, $\partial N$ denoting the boundary of $N$. But is this formally true?  I.e., is it true that

$$
\overline{C(M,\operatorname{int}(N))} = C(M,N)?
$$


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Prototype construction:  Let $N=[0,b)$ then any function $f$ can be approximated by:
$$
f_n= \min\left(\frac1{n},f\right)
,
$$
of course, these are continuous but not smooth (since we don't need smoothness this is not an issue).  I expect this type of construction can be generalized.

  [1]: https://en.wikipedia.org/wiki/Manifold#Manifold_with_boundary
  [2]: https://en.wikipedia.org/wiki/Compact_convergence