Epsilon and Delta

Date: 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/