Let's assume that $f:M\to N$, with $M\subseteq\mathbb{R}^m$, $N\subseteq\mathbb{R}^n$ is $k$-times differentiable on $M$ (which implies that $M$ is an open set).
In our material we have the statement that:
If you consider the restriction $f_{|M'}:= f:M'\to N$, where $M' \subseteq M$ and every point $m \in M'$ is a limit point of $M'$, then $f_{|M'}$ is also $k$-times differentiable.
I am wondering why it is sufficient that every point in $M'$ only has to be a limit point of $M'$ and not an interior point of $M'$?