Can you find a function that is differentiable at the origin but the function itself is not continuous at the origin?
(In reply to Possible Solution
That doesn't work, Gregg. If the function is not defined at the
origin, then I think that it is automatically not differentiable.
The problem in your posting is
lim (t->0) of 0/((x(x+t))t) is not equal to zero if x = 0,
because 0/((x(x+t))t) is undefined for all values of x, so the limit is also undefined.