# [SOLVED] Epsilon Delta Continuity

• Apr 30th 2010, 05:57 AM
ashishbaghudana
[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.

:)
• Apr 30th 2010, 06:43 AM
Plato
Hint: Given any real number $\displaystyle x_0$ in any $\displaystyle \delta-$neighborhood of that point there are two points say $\displaystyle a~\&~b$ such that $\displaystyle \left|f(a)-f(b)\right|=|1-0|>\frac{1}{2}$.
• Apr 30th 2010, 07:42 AM
ashishbaghudana
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?
• May 1st 2010, 03:15 AM
HallsofIvy
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.
• May 1st 2010, 06:10 AM
ashishbaghudana
Okay :) Thanks a lot!