Real AnalysisDate: 9/13/96 at 6:58:13 From: Anonymous Subject: Real Analysis Let I be a subset of R where I is an interval and let f map I to R be increasing on I. If c is not an end point of I, show that the jump of f at c is given by inf{f(y)-f(x): x<c<y, x, y is an element of I}. Date: 9/13/96 at 13:18:57 From: Doctor Tom Subject: Re: Real Analysis I can't help you on this because I don't know what you mean by "the jump of f at c". It's not a standard term (as far as I know), and if I were trying to make a definition, I would probably define it using exactly the words you did in your statement of what's to be proved. That is, I would have said: Definition: The "jump" of f at c is: inf{f(y)-f(x): x<c<y, x, y is an element of I. So if you can tell me the definition of "jump", I might be able to help. Or maybe you're asking why this makes sense as a definition? The reason is that an increasing function can have any countable number of discontinuities in the neighborhood of a point, so in any interval around c, the discontinuitites can be dense, so you have to use the inf definition. It simply says to try all combinations of values of f above and below c and find out how big the jump is between them. Then you find an infimum, or lower bound for those numbers. Obviously, you can't use any of the standard continuity arguments because there may be an infinite number of discontinuities in any interval around c. Anyway, if this doesn't help, write again, but tell me the "official" definition of "jump". -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 9/13/96 at 15:18:12 From: Barry Moore Subject: Re: Real Analysis The definition of "Jump" is : The Jump of f at c is defined to be (the right hand limit of f) minus (the left hand limit of F at), i.e. if f is an increasing function on[a,b] and c in an element of the interval (a,b). I hope this makes sense. Barry Date: 9/13/96 at 17:1:37 From: Doctor Tom Subject: Re: Real Analysis Okay. So one way to do this would be to show that there is a limit on both the right and left. Let me just look at the case where we approach c from below; the case from above will be basically the same, perhaps with some "greater-thans" replaced by "less-thans", and so on. Since f is increasing, if x < c, f(x) is always less than f(c), so all the f(x) values are bounded from above. If I look at smaller and smaller intervals below c, say between c and c-e, where e gets smaller and smaller, the range of f(x) on those intervals gets smaller and smaller. I claim that this range must condense to a point in the limit, because if the limit were a range, it has to contain at least 2 values that f takes on, and simply choose e small enough to chop out the lower one for a contradiction. So a limit exists from below (and a similar agrument shows that one exists from above). The jump is the difference of those values. I claim that the inf of your expression is the difference of those values. If your inf is bigger, I can find an "a" below c and a "b" above c so that f(b)-f(a) is arbitrarily close to your inf. Pick any smaller interval than from a to b, and any values in that interval will give f(b)-f(a) values smaller than what you claimed to be the inf (and closer to the difference of the limits), so the inf cannot be different from the difference of the limits (which is the jump). -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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