Question 72
Let $f : \mathbb{R} \to \mathbb{R}$ be a non-decreasing function and $F : \mathbb{R} \to \mathbb{R}$ a function having right and left finite derivatives at any point in $\mathbb{R}$ and $F(0) = 0$. Suppose that $\lim_{x \uparrow x_0} f(x) \le F'_s(x_0)$ and $\lim_{x \downarrow x_0} f(x) \ge F'_d(x_0)$, for any $x_0 \in \mathbb{R}$. Prove that $F(x) = \int_0^x f(t) \, dt$, $x \in \mathbb{R}$.