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?

>>15

The standard way to place an uncountable infinity of an shape on the plane is to "nest" slightly smaller copies of itself inside the original figure. Such that each copy corresponds to a shrinking or enlarging factor of a real number.

This works for the circle, and other shapes, because they have a certain property that I've heard referred to as "star shapes", but that's just a word we made up.

Basically a shape is a "Star shape" if there is a point inside a shape such that every point on the perimeter of the shape can be "seen" by that point. (ie. you can draw a straight line from this one point, to any point on the perimeter, without it having to cross another part of the perimeter)

It can easily be shown that being a star shape is sufficient to allow an uncountable infinity of them to be places on the plane, just shrink and enlarge w.r.t that point.

However I haven't seen a proof that it's necesary, but personally I think it is.

Now a figure of 8 isn't star shaped, the point inside has to be in one of the loops, and thus cannot "see" any of the perimeter points on the other loop.


This is not a proof that the figure of 8 can't be done. I'm merely trying to help >>15 understand why the converse of the proof in >>14 isn't true.

god this is a long post.