Originally Posted by

**dollydaggerxo** Hello

i need to show the following function is not continuous anywhere else but 0.

$\displaystyle f(x) = x^2 $if x is rational

$\displaystyle f(x) = 0$ if x is irrational

I have read the guide on epsilon delta proof and have managed to work out part of it, i have found that it is continuous at 0, and that is fine and completed.

I'm very stuck on the rest though. I tried to do this:

If a is a rational number there exists a set of numbers $\displaystyle y_1,y_2,...$ that converge to f(a).

So

$\displaystyle \lim_{y \rightarrow \infty} f(y) = \lim_{y \rightarrow \infty} y^2 = a^2$

If a is an irrational number, there exists a set of numbers $\displaystyle x_1,x_2,...$ that converge to f(a).

So

$\displaystyle \lim_{y \rightarrow \infty} f(x) = \lim_{y \rightarrow \infty} 0 = 0$

which are different, and therefore the function is not continuous when x is not 0.

Is this a valid proof?

Many Thanks