Mr Example.

I enjoyed those two Monsieur Ejemplé threads so I thought I'd pose a maths question of my own, in fact I'll do two on countability. They're not that hard, but I think they're more enjoyable than calculus

1. Let f : R -> R be monotonic.
Is  the set { x | f is discontinous at x} countable?
where x is in R.


2. A function f : N -> N is increaing if f(n)>= f(n+1)   (if it's bigger than OR equal to) and a decreasing function is similarly defined.

is the set {f | f is increasing} countable?
is the set {f | f is decreasing} countable?

Oh, that ones not very interesting.

basic point is every disc has a radius, say r, which is a real number.

Now by the theorem of archimedes I think, which states there is no upper bound to the natural numbers, there is a natural number n such that 1/n < r for every r in the reals.

Now, you can use this fact to show there must be a rational point within any disk with a real radius.

Basically a disc at centre (a,b) with radius r.

There exists a rational number between a and a +r/2, call this c1, and there exists a rational number between b ad b+r/2, call this c2.

Now (c1,c2) lies in the disk of radius r centered at (a,b)

However QxQ is countable because Q is countable, and as there is a rational point in every disc, there exists an injection from the number of disks in the plane and QxQ, thus the number or disks in countable.


It's a pretty simple idea, the rigourous prood is kinda hard to write down.