The derivative, at x= a, of f(x) is defined as
Further,
if and only if
for every sequence
converging to a.
Now, if there exist two sequences,
and
, converging to a, such that
then the function is NOT differentiable at a. That is, if the two sequences have different limits, then the limit as h goes to 0 cannot exist and so the function is differentiable. For any a other than 0, take
to be a sequence of rational numbers converging to a, and take
to be a sequence of irrational numbers converging to a.