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.
