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

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- Nov 9th 2012, 03:01 AMMichaelEngstlerNon 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 - Nov 9th 2012, 04:58 AMHallsofIvyRe: 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?

- Nov 9th 2012, 05:10 AMMichaelEngstlerRe: Non continuous function
Hey I just wrote which Dirichlet function I meant.

- Nov 9th 2012, 05:43 AMHallsofIvyRe: 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.

- Nov 9th 2012, 06:33 PMhollywoodRe: Non continuous function
Or look at the interval $\displaystyle (x-\delta,x+\delta)$ (using any value of $\displaystyle \delta$). Can you get $\displaystyle |f(x)-f(x_0)|<\epsilon$?

- Hollywood