Friday, 23 August 2019

calculus - Show a function whose derivative is bounded is also bounded in an interval



Suppose g is differentiable over (a,b] (i.e. g is defined and differentiable over (a,c), where (a,c)(a,b]), and |g'(p)| M (M is a real number) for all p in (a,b]. Prove that |g(p)|Q for some real number over (a,b].




I looked at a similar solution here: prove that a function whose derivative is bounded also bounded
but I'm not sure if they are asking the same thing, and I'm having trouble figuring out the case when x(a,x0) (x here is point p). Could someone give a complete proof of this problem?


Answer



Use mean value theorem to get g(x)=(xb)g(c)+g(b). If x is bounded then clearly the RHS of the equation is bounded.


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