# Proving Continuity of a function

• Feb 28th 2009, 04:03 PM
redsoxfan325
Proving Continuity of a function
f(x)={0 if x is irrational | 1/n if x = m/n ∈ Q}

I need to prove that f is continuous at every irrational point and that f has a simple discontinuity at every rational point.

I have no idea where to begin because for all irrational x, there will always be a rational number in an arbitrarily small neighborhood around it.

You need to show that $\displaystyle \lim_{x\rightarrow a} f(x)= 0$ for all a. Yes, inside any interval around an irrational (or rational) number there exist a rational number. But, since there are only a finite number of m such that m/n is in such an interval, as the interval gets smaller, n must get larger and larger.
You need to show that $\displaystyle \lim_{x\rightarrow a} f(x)= 0$ for all a. Yes, inside any interval around an irrational (or rational) number there exist a rational number. But, since there are only a finite number of m such that m/n is in such an interval, as the interval gets smaller, n must get larger and larger.
How do I know that (the bolded part)? Because, I agree, if I can prove n gets larger and larger, then I'll be able to prove continuity. My worry was that the irrational point would contain some simple fraction like 1/2 in its neighborhood. How can I prove that $\displaystyle n \rightarrow \infty$ as $\displaystyle \frac{m}{n} \rightarrow x$?