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?

>>22
It isn't, I'm wrong. It would be an infinite binary tree, and infinite binary trees has an uncountable infinite number of paths, but only a countable infinite number of nodes. Beginners' mistake, really.

>>24
No, he's only requiring that if an inlaid shape shares area with one of the two loop of an outer shape, it may not also share area with the other loop.