[SOLVED] continuous at irrational points 1. The problem statement, all variables and given/known data Every rational x can be written in the form m/n where n>0, and m and n are integers without any common divisors. When x = 0, we take n=1. Consider the function f defined on the reals by f(x) = 0 if x is irrational and f(x) = 1/n if x = m/n Prove that f is continuous at every irrational point, and that the right and left-hand limits of f exist at every rational point. 2. Relevant equations 3. The attempt at a solution If x is irrational, then when you get closer and closer to it, it will get harder and harder to express it is a rational number and you will need larger and larger n to do it. I need to make that precise somehow. For the second part, I am guessing that the right and left-hand limits will always be zero. Anything else would be kind of weird. And that is basically for the same reason I gave above, any rational sequence converging to a rational number will get "nastier and nastier" as it gets closer. But I need to make that precise somehow.