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?

>>21

OP here, well then it looks like we have ourselves a contradiction.


By the same reasoning as the disk argument, each loop on a figure of eight contains a rational point.

Therefore you can assign each figure of eight a pair of rational points, one in each loop, that cannot also be assigned to any other loop.


Thus the number of figures of eight can be injected into (QxQ)x(QxQ), which is countable.


The other argument is interesting, but I'm not entirely sure if it's correct.