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Mr Example.

Name: Anonymous 2007-08-30 13:43 ID:eUFPtE+p

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?

Name: !Ep8pui8Vw2 2007-08-30 14:35 ID:1IVSPEHh

Haha, is this the Trinity guy again? Fucking hell.

1: Each "interval" it jumps by at a discontinuity contains some point in Q. This is quite easy to prove, but tedious to write out.

2:
a) Assume increasing functions countable, list as f1, f2, f3... Go through and define a function so that it's not on the list (I'm in a hurry actually, I can do this more formally later tonight maybe). Hence uncountable.

3: Countable, and of course you can have a (not strictly) decreasing function on the natural numbers. The proof will involve there being a countable number of permutations of numbers below whatever f(1) is, then countable union of countable sets arguments.

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