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Topic: Countable Discontinuities (Read 3889 times) |
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william wu
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Countable Discontinuities
« on: Apr 21st, 2004, 9:07pm » |
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Let f be a monotonic increasing function that maps from the reals to the reals. Prove that f can have at most a countable number of discontinuities.
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Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
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Re: Countable Discontinuities
« Reply #1 on: Apr 22nd, 2004, 4:16pm » |
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Proof: Since f is monotonic, f(a-) = sup{f(x) : x<a} and f(a+) = inf{f(x) : x > a}, so both the right and left limits exist at every point. If a < b are two discontinuities of f, then f(a+) < f(b-), since f is increasing. Hence the intervals (f(a-), f(a+)) and (f(b-), f(b+)) are disjoint. Thus every discontinuity of f has a unique interval associated with it, disjoint from the intervals of other discontinuities. But each such interval must contain a rational number. Hence there can only be countably many.
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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Barukh
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Re: Countable Discontinuities
« Reply #2 on: Apr 23rd, 2004, 11:06am » |
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Bravo, Icarus! I liked your argument very much!
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Icarus
wu::riddles Moderator Uberpuzzler
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Re: Countable Discontinuities
« Reply #3 on: Apr 23rd, 2004, 4:16pm » |
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I did leave a minor point out: Because f is increasing, f(x-) <= f(x) <= f(x+). So if f(x-) = f(x+), they must both equal f(x), and f is thus continuous at x. Hence for any discontinuity a, f(a-) < f(a+), and the interval between them is not empty. Let me add this follow-up challenge: If any arbitrary Real function f possesses only jump discontinuities (ie, f(a-) and f(a+) exist but are not equal), is it possible to have uncountably many of them?
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« Last Edit: Apr 23rd, 2004, 5:02pm by Icarus » |
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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Eigenray
wu::riddles Moderator Uberpuzzler
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Re: Countable Discontinuities
« Reply #4 on: May 2nd, 2004, 11:33pm » |
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on Apr 23rd, 2004, 4:16pm, Icarus wrote: Let me add this follow-up challenge: If any arbitrary Real function f possesses only jump discontinuities (ie, f(a-) and f(a+) exist but are not equal), is it possible to have uncountably many of them? |
| For each x [in] [bbr] define tx = |f(x+)-f(x-)|. We show that each Sn := {x | tx > 1/n} is countable, and so, therefore, is their union, [cup]n=1[infty]Sn = {x | tx > 0}. Fix n. For each x [in] Sn, the limit f(x-) exists, so there exists x' < x such that |f(y)-f(x-)| < 1/2n whenever x' < y < x. For any such y, it follows that |f(y+)-f(x-)| and |f(y-)-f(x-)| are both [le] 1/2n, so that ty = |f(y+)-f(y-)| [le] 1/n, which means y [notin] Sn. It follows that the intervals (x',x) are disjoint for distinct values of x [in] Sn, so there are only countably many (each contains a rational).
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Icarus
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Re: Countable Discontinuities
« Reply #5 on: May 4th, 2004, 6:04pm » |
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Excellent proof!
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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous
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