This function is continuous only at the point x=0. Why? Well, it's certainly not continuous at any other point, since if , then by taking a sequence of rational numbers converging to and then a sequence of irrational numbers converging to , you can see that doesn't exist. But why is f continuous at 0? We know that f(0)=0 from the definition. We also have that for all x (since f is always either x or -x). Since , it follows from the pinching (or sandwich) theorem that . Since this is f(0), this means that f is continuous at 0.