It is
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.