RO-NMO 2016 Problem 123

Let be a real number $a,$ and a nondecreasing function $f:\mathbb{R}\longrightarrow\mathbb{R} .$ Prove that $f$ is continuous in $a$ if and only if there exists a sequence $\left( a_n \right)_{n\ge 1}$ of real positive numbers such that $$\int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} ,$$ for all natural numbers $n.$ Dan Marinescu