RO-NMO 2011 Problem 124

Let $f,F:\mathbb{R}\longrightarrow\mathbb{R}$ be two functions such that $f$ is nondecreasing, $F$ admits finite lateral derivates in every point of its domain, $$\lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} ,$$ for all real numbers $y,$ and $F(0)=0.$ Prove that $F(x)=\int_0^x f(t)dt,$ for all real numbers $x.$