Can you find a function that is differentiable at the origin but the function itself is not continuous at the origin?
How about f(x)=x/x?
Since 0/0 is not defined f(x) is not continous at x=0.
But, f'(x)=lim (t->0) of (((x+t)/(x+t)) - (x/x))/t
=lim (t->0) of ((x+t)x - x(x+t))/((x(x+t))t)
=lim (t->0) of 0/((x(x+t))t)=0
So, f'(x)=0 for all defined x including x=0.
Posted by gregg
on 2006-10-25 00:23:28