Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold *with boundary*. 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.

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

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

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.