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: For continuous functions... by Steve Herman)
Okay good point.
What about adding a step where if the integrals over a particular range for two functions f and g are equivalent we compare the integrals of |f(x)|dx and |g(x)|dx over the same range?
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Posted by Eric
on 2006-08-23 17:26:45 |
re(2): For continuous functions...
