Answer
$f(x)= c \cdot x$ is continuous on $R^n$ when $c \in V_n$
Work Step by Step
Let us consider $c=(c_1,c_2,....,c_n)$ and $x=(x_1,x_2,...,x_n)$ ...(1)
From the question, let us consider $f(x)= c \cdot x$ and $a=a_1,a_2,....a_n$
From equation (1), we have
$\lim\limits_{x \to a}f(x)f(x)= \lim\limits_{x \to a}(c_1,c_2,....,c_n) \cdot \lim\limits_{x \to a}(x_1,x_2,...,x_n)$
$=c_1a_1+.....+c_na_n$
Therefore, $\lim\limits_{x \to a}f(x)=f(a)$
This shows that $f(x)= c \cdot x$ is continuous on $R^n$ when $c \in V_n$
