
Gradient
Consider the function f defined by
1) f(x,y) = 0 unless x>0 and $\displaystyle x^2 < y < 3x^2$
2) for each x > 0$\displaystyle f(x, 2x^2) = x$
3) $\displaystyle 0 \leq f(x,y) \leq x$ for all (x,y) with x>0
Modify this to get a function g with
$\displaystyle g_1 (0,0) =g_2 (0,0) = 1$
yet there is no direction of maximal change.
I know f has no direction of maximal change because$\displaystyle f_1 (0,0)$and$\displaystyle f_2 (0,0) $are both 0, and so the gradient is zero and the angle between the gradient and any direction vector is undefined. How can this happen when the gradient is 1?