Question

f is a function define as f(x)=x^2 for x rational and f(x)=0 for x irrational show f is differentiable at x=0

Answer 1

First, let's try to get an intuitive idea of why this function should be continuous at this point and not other points. Basically, it is because the rational x's are approaching 0, and the irrational x's are already there. That's why it should be differentiable at zero and nowhere else.

Set up the definition of the derivative in your favorite notation and then use the delta epsilon definition of limit to prove the limit exists. The limit does exist and is equal to O, so the function is differentiable at this point.