# Math Help - [SOLVED] Epsilon Delta Continuity

1. ## [SOLVED] Epsilon Delta Continuity

Hi

I have a question based on the epsilon delta definition of continuity.

f(x) = 1, when x is rational and 0 when x is irrational.

Prove that f is discontinuous at every point.

2. Hint: Given any real number $x_0$ in any $\delta-$neighborhood of that point there are two points say $a~\&~b$ such that $\left|f(a)-f(b)\right|=|1-0|>\frac{1}{2}$.

3. Thanks. So this means in the delta band, however small it may be, I can find 2 real numbers such one is a rational number and the other an irrational number?

4. Yes, that is a fundamental property of the rational and irrational numbers- they are both "dense" in the real numbers. In any interval of real numbers, no matter how small, there exist both rational and irrational numbers. Whoever gave you this problem expected you to know that.

5. Okay Thanks a lot!