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 May 1st, 2007, 02:16 AM #1 Site Founder     Joined: Nov 2006 From: France Posts: 824 Thanks: 7 Bounded derivatives Here's a cute problem. Let f: R->R be a mapping which is N+1 times differentiable, and such that both f and f^(N+1) (the N+1-th derivative of f) are bounded on R. Show that all derivatives f^(k) of f are bounded on R. R designates the field of all real numbers.
 October 30th, 2007, 09:57 AM #2 Newbie   Joined: Oct 2007 From: Brazil Posts: 8 Thanks: 0 You're assuming f^k exists on R for every k, right? This is not part of the conclusion, right? If, for example, n =1, then the fact that f is twice differentiable and f and f'' are bounded on R does not impliy f has derivatives of all orders on R. I'm confused here. Artur
 October 30th, 2007, 10:47 AM #3 Site Founder     Joined: Nov 2006 From: France Posts: 824 Thanks: 7 No, what I meant is to prove that given the hypotheses, the result holds for every k<=N+1. Of course, f does not end up being infinitely differentiable.
 October 30th, 2007, 11:48 AM #4 Senior Member   Joined: Oct 2007 From: France Posts: 121 Thanks: 1 Can we suppose that n=N?
 October 30th, 2007, 01:47 PM #5 Site Founder     Joined: Nov 2006 From: France Posts: 824 Thanks: 7 Sorry, I edited the original message. Actually there's only N.
 November 1st, 2007, 09:35 AM #6 Newbie   Joined: Oct 2007 Posts: 1 Thanks: 0 Well it is true for N=1... thats all i can show for now, dunno how to expand it to N=n. First if f is N+1 times differentialable on R f^k is continous everywhere. so if any derivate is unbounded its at x>>1 okay... if f is bounded then f(x)f(a)-f(b)=f'(c)(b-a) for some c contained in (a,b) {meanvaluetheorem continous functions} if f' is unbounded, then given any £ > 0 there exists a real number d in R such that f'(d) > 1/£^2 Now consider the interval (d-£,d+£) by i) there exists a q contained in the interval where f'(q)<2M*2£ Meanvalue theorem applied again yields f'(d)-f(q)=f''(k)*(q-d) for some k contained in (q,d) or (d,q) who have length < £ so f''(k)=(f'(d)-f(q))/(q-d)> (1/£^2 - 2M*2£) / £ which gets arbitrary large ie we have shown the existance of a such k for every chosen £, thus f''(k) must be unbounded (contradiction, so f' is unbounded is false)

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