Find a function f:R->R (R the set of real numbers), such that
1. f has a discontinuity in every rational number, but is continous everywhere else, and
2. f is monotonic: x<y → f(x)<f(y)
Note: Textbooks frequently present examples of functions that meet only the first condition; requiring monotonicity makes for a slightly more challenging problem.
(In reply to
re: Oops... by Steve Herman)
In fact there is a solution and you'll be surprised how simple it is, once you found it. You and Ken's observations made me think a bit more about this problem and I am realizing that my previous hint was maybe overly complicated, although the w-function is quite interesting to study, don't you think?
So here is another hint if you care for it:
1. Given a rational number r, what is the simplest function f_r you can think of, that is discontinuous at r and continuous elsewhere?
2. Now imagine that you have a whole bunch of rational numbers, how could you use the corresponding f_r's to construct a function that is discontinuous at all your rational numbers?
3. Can that be extended if the "whole bunch" consists of ALL rational numbers?
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Posted by JLo
on 2006-08-20 10:42:45 |
re(2): Oops... Maybe another hint?
