Define *f*(*x*,*y*) to be . Then for any unit vector , the directional derivative at the
origin is by definition

In particular, and (since these are directional
derivatives for and respectively). If *f* were
differentiable at the origin, then would equal for every . But
is not always zero, so *f* is not
differentiable.

Here's the graph of *f*:

Mon May 5 12:53:33 CDT 1997