How to plot a radial cutoff functions

Question

I want to plot the smooth positive radial cutoff function $\chi_2$ defined by $$ \chi_2(x,y) =1 \qquad(100|x-y| \le |y|), $$ $$ \chi_2(x,y)=0 \qquad(|y|\le 50|x-y|) $$ and also the function $\chi_1 = 1 - \chi_2$. I hope that I can plot this in mathematica, but it seems too hard for me.

Answer 1

Your function is neither welldefined for $(x,y)=(0,0)$ nor defined for all $(x,y)$. For instance for $(x,y)=(5151,5075)$ you get $$ 100|x-y|=100\cdot76=7600>5075=|y| $$ and $$ |y|=5075>3800=50\cdot76=50|x-y| $$