A more sophisticated example than the last one is the following. Define *f*(*x*)
by

Now, *f*(*x*) is continuous at *x*=0, since as , , so . *f* also has a
derivative at *x*=0:

which you can show goes to zero by using L'Hôpital's rule on
. In fact, using mathematical
induction and a considerable amount of work, you can show that all of *f*'s
derivatives exist at 0 and are equal to zero.

This says that *f* has a very nice Taylor expansion around the origin. Since
for all *n*, the Taylor series around the origin is simply a sum
of zeroes, so it is identically zero. However, this equals *f*(*x*) only at the
point *x*=0! Here's a picture of *f*. Notice that *f* becomes very flat at the
origin. However, it only equals 0 when *x*=0, since *e* raised to any finite
power is positive.

Mon May 5 12:53:33 CDT 1997