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 is seems that every continuous function $f:M\rightarrow N$ should be able to be approximated by continuous functions of the form $g:M\rightarrow int(N)$; where $N=int(N)\cup \partial N$, $\partial N$ denotes the boundary of $N$ and $int(N)$ the $n$-dimensional manifold defined by removing $N's$ boundary points but is this formally true? ie: is:

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