Recent content by Kkathy

So, qx contains all the q < x. And y > x. When x is rational, x will be contained in qy because qy is a set of rational numbers less than y and x is a rational number less than y. However, x cannot be the upper bound of y because the numbers are dense, so there will always have to be a number...

I see your problem, let me try again. Let x be rational. Choose y∈R such that y>x. Let qx be the set of rationals < x and qy be the set of rationals < y. Then x=qm for some m (since x is rational, it is part of an enumeration of rationals) and qm∈qy. Now f(y)=Ʃn∈qy(1/n2)≥f(x)+1/m2. Then...

Okay, I think I can see it now. From the beginning: Let x and y be rational, and choose y>x. Let qx be the set of rationals < x and qy be the set of rationals < y. Then x=qm for some m (since x is rational, it is part of an enumeration of rationals) and qm∈qy. Now...

For any y and x where y>x, then f(y)-f(x) = Ʃx<q<y(1/n2). Because the set of x<qn<y is a finite set of numbers, then as (y-x)→0, f(y)-f(x)→0.

I can see that if I pick two points y and x, then as y and x get closer then f(y) and f(x) will get closer. I think the reason that irrational numbers are continuous is because the are not dense in the set of rational numbers. Rational numbers are dense so it won't be possible to choose a delta...

Part of the problem I'm having here is I don't understand what f(x) is. For example, how would I calculate f(1)? The index of summation would be qn < 1 but what is that, is it a finite set of numbers? How about for an irrational number like f(2/3)?

I don't think I did a good enough job of describing f(x): f(x)=∑qn<x(1/n2) So, the index of summation is qn<x, where the series of q is an enumeration of the rational numbers. What is being summed is one over n-squared. The goal is to prove that f is continuous at each irrational and...

Real Analysis--Prove Continuous at each irrational and discontinuous at each rational The question is, Let {q1, q2...qn} be an enumeration of the rational numbers. Consider the function f(x)=Summation(1/n^2). Prove that f is continuous at each rational and discontinuous at each irrational...