1. ## Non continuous function

Hey everyone,
How would you prove that this function is not continuous in every x != 1:
x-1 +((x-1)^2)*D(x)

D(x) is Dirichlet function.
if x rational -> D(x) = 0
if not -> D(x) = 1

Thanks a lot,
Michael Engslter

2. ## Re: Non continuous function

There are several different "Dirichlet functions" but I assume you mean the most common, D(x)= 1 if x is rational, 0 otherwise. Given any x, there exist a sequence of rational numbers that converges to it and there exist a sequence of irrational numbers that converges to it. What is the limit of this function as you approach x along either sequence?

3. ## Re: Non continuous function

Hey I just wrote which Dirichlet function I meant.

4. ## Re: Non continuous function

Okay, so what I said still applies. Take the limits as you approach x along a sequence of rational numbers and along a sequence of irrational numbers.

5. ## Re: Non continuous function

Or look at the interval $(x-\delta,x+\delta)$ (using any value of $\delta$). Can you get $|f(x)-f(x_0)|<\epsilon$?

- Hollywood