A function $h : A → \mathbb{R}$ is Lipschitz continuous if $\exists K$ s.t.
$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$
Suppose that $I = [a, b]$ is a closed, bounded interval; and $g : I → \mathbb{R}$ is differentiable on $I$ and the function $G = Dg = g' : I → \mathbb{R}$ is continuous. Prove that $g$ is Lipschitz continuous on $I$.