Given two normed vector spaces $(V,\|\cdot\|_V),(W,\|\cdot\|_W)$ over some field $F$ equipped with some modulus $|\cdot|$, and a function $f:V \to W$, we say $f$ is differentiable at ${\bf a}\in V$ along direction ${\bf{v}}\in V$ if
$\lim_{h→0}\frac{‖f({\bf a}+h{\bf v})-f({\bf a})-h{\bf u}‖_W}{|h|}=0$
where the norms and modulus are real-valued, and $h \to 0$ is a notation for $|h| \to 0$. We say
$\frac{\partial f({\bf a})}{\partial {\bf v}}={\bf u}$
is the derivative of $f$ along direction $\bf {v}$ at $\bf a$. In particular, if $\|{\bf v}\|_V=1$, then such defined $\frac{\partial f({\bf a})}{\partial {\bf v}}$ is the directional derivative of $f$ at $\bf a$ with respect to direction $\bf v$.
Now the question is to prove
if $f$ is differentiable near $\bf a$ along any direction and $f$ is continuous at $\bf a$, then $g({\bf v})=\frac{\partial f({\bf a})}{\partial {\bf v}}$ is a linear map, i.e.
1) $\frac{\partial f({\bf a})}{\partial {c\bf v}}=c\frac{\partial f({\bf a})}{\partial {\bf v}}$
2)$\frac{\partial f({\bf a})}{\partial {\bf (u+v)}}=\frac{\partial f({\bf a})}{\partial {\bf u}}+\frac{\partial f({\bf a})}{\partial {\bf v}}$
for any $c\in F$,${\bf u},{\bf v} \in V$
I am able to show the first by the following,
but I am not able to solve the second one. Thanks!