Let $f:\mathbb{R}^n\to\mathbb{R}$ be the linear mapping.
a) Is $f$ uniformly continuous on $\mathbb{R}^n$?
b) Is $f$ differentiable on $\mathbb{R}^n$?
c) Find the general form of $f$.
My attempt for part a)
Let $(e_1, e_2,...,e_n)$ be the standard basis of $\mathbb{R}^n$.For $x\in\mathbb{R}^n$, $x=\sum\limits_{i=1}^{n}x_ie_i$.
Hence,$$|f(x)|=|f(\sum\limits_{i=1}^{n}x_ie_i)|=|\sum\limits_{i=1}^{n} x_if(e_i)|\leq \sum\limits_{i=1}^{n}|x_if(e_i)|\leq\max\limits_{1\leq i\leq n}|f(e_i)|\sum\limits_{i=1}^{n}|x_i|\leq \max\limits_{1\leq i\leq n}|f(e_i)|\sqrt{n\sum\limits_{i=1}^{n}|x_i|^2}=(\max\limits_{1\leq i\leq n}|f(e_i)|\sqrt{n})\sqrt{\sum\limits_{i=1}^{n}|x_i|^2}=M||x||$$ with $M=\max\limits_{1\leq i\leq n}|f(e_i)|\sqrt{n}$ is positively constant (Here, I chose the Euclidean norm in $\mathbb{R}^n$).
So, $|f(x)-f(y)|=|f(x-y)|\leq M||x-y||\quad\forall x,y\in\mathbb{R}^n$, which implies $f$ is uniformly continuous.
Is my proof for part a) valid?
Could someone help me with part b) and c)? I would very much appreciate if someone could shed light on these parts.