| Nov. 15th, 2009 @ 10:57 am Couple of Quickies. |
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First, trying to find an explicit map from (0,1) to R, which takes rationals to rationals, and irrationals to irrationals. I did find one from (0,1) to [0,1] (which was part two), but not seeing a way of extending that. (Took Q intersect (0,1), which is countable. Enumerated it {r1, r2, r3, ...} and defined f:[0,1] -> (0,1) by f(0) = r1, f(1) = r2, f(rn) = r(n+2), and f(x) = x else.)
Secondly;
I'm trying to understand this proof of "An absolutely continuous function is of bounded variation on some interval [a,b]"
In the proof given by Royden, we take ε = 1, and then take δ that corresponds to it. We then take the finite partition that correlates to this, and increase the number of intervals (for absolute continuity) to K intervals with total length less than δ, and K is the largest integer less than 1 - (b-a)/δ. This choice of K is where I'm sticking. Because then we have t, given as the finite sum from 1 to n of |f(x_i) - f(x_(i-1))|. Now, I see that this sum is the same for absolute continuity, so t < 1, by Absolute continuity. But Royden then says t <= K, and thus the suprema of all t is T <= K. I get that. It's the choice of K.
Thanks! |
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