Define *f*(*x*,*y*) by

Along any line through the origin, *f* approaches zero. Indeed, any line through
(0,0) may be parametrized by *x*=*ta*, *y*=*tb* for some constants *a* and *b*,
not both zero. As *t* goes to 0, ,
which goes to 0 as *t* goes to zero for any *a* and *b*. However, does not exist: if you approach the origin along the
parabola , you note that is constantly 1/2, which doesn't go
to zero.

Here's a picture, where we're looking along the positive *x* axis towards the
origin. What happens as you approach along any ray which is above the *x* axis
is that you go up the ridge which is above and then slide down the other
side as you hit the origin. However, if you come in on , you stay on top
of the ridge the whole way, and get a limit of 1/2. (If you come in on a ray
below the *x* axis, you go down into the trough which is below and then
up on the other side until you hit the origin.)

Mon May 5 12:53:33 CDT 1997