We had a problem in class today which read
f(x) = x^2 if xЭQ
f(x) = 0 otherwise.
We are studying differentiability, and my tutor said that this function was differentiable at 0 but not at any other point.
I can see how to show its differentiable at 0, but I dont know how I would go about proving that is isn't differentiable at any other point.
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.