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?

>>14
Ah, I see. (Strictly speaking,  natural number n such that 1/n < r for every r in the positive reals.)

Then that works pretty much directly for filled 8-shapes, assuming they can't be infinitely thin, but doesn't work for outlines, since any rational point can be shared by (countably) infinitely many nested 8-shapes, right?
But if that is the case, you can place an uncountable infinite number of 8-shapes, so I must have misunderstood something.