f: R^2 -> R f(x,y) = |x| + |y|
(a) Find all directional derivates at (0,0) in the direction of u not equal to zero if they exist. And evaluate when they do.
(b) Do the partial derivatives exist at (0,0)?
(c) Is it differentiable at (0,0)?
(d)Is it continuous at (0,0)?
(a) No they do not exist. I have:
f'(0;u) = lim [t->0] 1/t[f((0,0) + t(h,k)) - f(0,0)] = lim [t->0] 1/t(|th| + |tk|)
Which then equals |h| + |k| and -|h| - |k| for t > 0 and t < 0 respectively. So the "left and right" limits aren't equal. Therefore, the limit cannot exist.
(b) No, because the directional derivatives don't.
(c) No it is not differentiable. Since all directional deritaves at 0 don't exist it implies it is not differentiable at 0.
(d) Yes it is continuous at 0.
lim [ (x,y) -> (0,0) ] |x| = 0 and lim [ (x,y) -> (0,0) ] |y| = 0
These are both proved by setting delta = epsilon. And therefore the summation of the two exists, so we have:
lim [ (x,y) -> (0,0) ] f(x,y) = lim [ (x,y) -> (0,0) ] |x| + |y| = lim [ (x,y) -> (0,0) ] |x| + lim [ (x,y) -> (0,0) ] |y| = 0 + 0 = 0.
Can someone verify whether I'm right or wrong on any of the parts? I'd really appreciate it!!