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)$\supset$(a,b]), and |g'(p)|$\le$ M (M is a real number) for all p in (a,b]. Prove that |g(p)|$\le$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 $x\in(a,x_0)$ (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.