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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.

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# Homework Help: Continuous at irrational points

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