Sort the set of functions f:R→R, with R being the set of real numbers. This means you have to find an
order "«" that lets you compare any two pairs of unequal functions f and g; unequal means, f(x)≠g(x) for at least one x.
More precisely, these are the requirements for the order "«" you are challenged to find:
1. If f≠g, either f«g or g«f.
2. If f«g and g«h then f«h
You might be tempted to declare f«g when f(x)<g(x) for all x but that would of course fail because e.g. f(x)=x and g(x)=-x would not be comparable with respect to your order.
For a much, much easier challenge, start by finding an order for all continuous functions.
(In reply to
re(2): Still in Deep Water by Richard)
While I am pretty good at counterexamples, I don't have an approach yet
to sorting continuous functions, which JLo says is "the much, much
easier challenge". Personally, I feel like I'm being taunted.
This thread is still young in "JLo-years", so I haven't given up hope that somebody will come up with something.
re(3): Still in Deep Water
