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.