(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if y>=x>=0:

a) [tex] y^2 arctan y - x^2 arctan x >= (y^2 - x^2) arctan x [/tex]

b) [tex] \ | \ y^2 arctan y - x^2 arctan x \ | \ >= (y^2 - x^2) arctan x [/tex]

c) use (b) to prove that x^2 arctan(x) isn't UC in R.

2. Relevant equations

3. The attempt at a solution

a) We have to prove that [tex] y^2 ( arctan y - arctan x ) >= 0 [/tex]

And since arctan(y) - arctan(x) >= 0 for all y>=x and y^2 > 0 this is true.

b)Since [tex] y^2 arctan y - x^2 arctan x >= 0 [/tex] for all y>=x>=0 then this is obviously true from a.

c)If we choose Epsilon (E) = 1/2 , Lambda (L) > 0 and y = x+L then

(y^2 - x^2)arctan(x) = (2xL - L^2)arctanx and the limit of that at infinity is infinity. So we can find N>0 so that for every x>N

(y^2 - x^2)arctan(x) = |y^2 arctan(y) - x^2 arctan(x)| > E

Now my questions are:

1) a & b seemed too trivial -are those the right are answers?

2)Is (c) right?

Thanks.

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# Homework Help: Uniform continuity

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