Let f be defined on [0,1] by the formula: f(x) = 1 (if x is rational) and f(x) = 0 if x is irrational. Prove that f is continuous at no point in the interval [0,1].

So I figured proof by contradiction would work best for this problem. So assume that f is continuous at some point p in the interval [0,1]. I'm not sure how to proceed from there though.

Question

Let f be defined on [0,1] by the formula: f(x) = 1 (if x is rational) and f(x) = 0 if x is irrational. Prove that f is continuous at no point in the interval [0,1]. 

So I figured proof by contradiction would work best for this problem. So assume that f is continuous at some point p in the interval [0,1]. I'm not sure how to proceed from there though.

dumbcow · Answer

https://en.wikipedia.org/wiki/Nowhere_continuous_function