Let $f: \mathbb{R}^2\to \mathbb{R}$ be defined by setting $f(x,y)=|x|+|y|$
(a) For which vectors $u\ne 0$ does $f'(0; u)$ exist? Evaluate it when it exists.
(b) Do $D_1f$ and $D_2f$ exist at $0$?
(c) Is $f$ differentiable at $0$?
(d) Is $f$ continuous at $0$?
I have thought a lot about this problem:
For (a), be $u\neq 0, u:=(h,k)$, then $\lim_{t\to 0}\frac{f(0+tu)-f(0)}{t}=\lim_{t\to 0}\frac{f(th,tk)}{t}=\lim_{t\to 0}\frac{|th|+|tk|}{t}=(|h|+|k|)\lim_{t\to 0}\frac{|t|}{t}$ and just as $\lim_{t\to 0}\frac{|t|}{t}$ does not exist, necessarily $h=k=0$, with which no directional derivative exists, this immediately tells us that in (b) neither $D_1f$ and $D_2f$ exist and that $f$ is not differentiable in $(0,0)$. Also clearly $f$ is continuous in $(0,0)$. Is this reasoning correct? Could someone help me by giving me suggestions? Thank you.