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: Still in Deep Water by Steve Herman)

I agree that your counterexample shoots down my method. Just goes to show you what I don't know about continuity about a point.

Perhaps my method would work for functions that have a continuous derivative everywhere, or for continuous functions which are locally of bounded variation.

Edited on August 24, 2006, 3:51 am

Posted by Richard on 2006-08-24 02:24:49