Throughout this section we assume $V$ and $W$ are normed vector spaces, with $V'$ an open subset of $V$ and $f$ a function defined $V'\to W.$
Definition: The function $f:V'\to W$ is differentiable at $a\in V'\subseteq V$ if there exists a continuous and linear transformation $L:V\rightarrow W$ such that $$\lim_{h\rightarrow 0}\frac{|f(a+h)-f(a)-L(h)|_W}{|h|_V}=0.$$ The linear transformation $L$ is called the derivative of $f$ at $a$ and it's denoted by $$f'_a = f'(a) = Df(a) = D_a f.$$
My question: for the limit above to be well defined we must have that $0$ is a limit point of the function $$h\mapsto \frac{|f(a+h)-f(a)-L(h)|_W}{|h|_V}$$ which is -I believe- equivalent to $a$ being a limit point of $V'$. Should the definition not require that $a$ to be a limit point of $V'$ then?