Epsilon and DeltaDate: 04/18/99 at 02:39:49 From: Brandi Holcomb Subject: College Advanced Calculus How can I show that sin(x^2) is not uniformly continuous on the reals? (I.e. what are my epsilon and delta equal to?) Date: 04/18/99 at 05:49:59 From: Doctor Mitteldorf Subject: Re: College Advanced Calculus Dear Brandi, You have to go back to the definitions and see exactly what it is you're being asked to prove. This may seem like a step backwards, and a waste of time, but by the time you understand the definitions, the proof will be easy. "Continuity" is just saying that - First we agree on the spot (x,y) on the curve - Then I come up with an epsilon - Then you have to come up with a safe delta so that you can assure me the function doesn't vary by more than that amount within a region around (x,y) of size delta. "uniform continuity" is a bit stronger: - First I come up with an epsilon - Then you have to come up with a safe delta - Then I get to pick the (x,y) where we apply it. You still must assure me the function doesn't vary by more than my epsilon as long as x stays within a region around (x,y) of size delta. But since I get to choose (x,y) I can see how small your delta was, then choose a place where sin(x^2) gyrates so fast that it violates your guarantee. I can do this because the larger x is, the faster the sin function oscillates. In other words, x^2 is increasing more and more rapidly the farther out I go, so sin(x^2) goes through tighter and tighter cycles with increasing x. Can you see from this how to write up your proof? - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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