You need to show that for any we can find so that there is a point such that . To show this think along the x-axis. Therefore, there is no way to define the function to even a continous function at .Nevertheless, f is not smooth as a function of the pair (x,y).
b.) Show this in the case N = 2 by demonstrating that the function is not even bounded in the neighborhood of the origin (0,0).
Show does not exist.c.) Show this in the case N = 1 by demonstrating that the function though bounded is not actually continuous as a function of (x,y).
Defining will make the function continous. It is continous for if . Thus, the function is continous. To show it is not differenciable it is sufficient to show is not differenciable at (this is along the path ).d.) Show this in the case N = by showing that though the function is now continuous, it is not smooth along the line x = y.