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)=(x−b)g′(c)+g(b). If x is bounded then clearly the RHS of the equation is bounded.
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