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

1) f is some function who has a bounded derivative in (a,b). In other words, there's some M>0 so that |f'(x)|<M for all x in (a,b).

Prove that f is bounded in (a,b).

2) f has a bounded second derivative in (a,b), prove that f in uniformly continues in (a,b).

2. Relevant equations

3. The attempt at a solution

1) For all x in (a,b), (f(x)-f(a))/(x-a) = f'(c) =>

|f(x)-f(a)| = |f'(c)||x-a| =< M|b-a| (c is in (a,x))

and so

f(a) - M|b-a| =< f(x) >= f(a) + M|b-a| and so f(x) is bounded in (a,b).

2)Using (1) we see that f'(x) is bounded in (a,b). So there's some M>0 so that |f'(x)|<=M. And so for all x,y in (a,b)

(f(x)-f(y))/(x-y) = f'(c) =< |M| and so

|f(x)-f(y)| =< |x-y|M and it's easy to see that f is UC in (a,b).

Did I do that right? It some what bothers me that in (1) I also could have proved that f is UC which is a stronger result than the fact that it's bounded the same way that I proved it in (2). Is that right? If so then why did they only ask to prove that it was bounded?

Thanks.

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# Homework Help: Uniform continuity and derivatives

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