Question 67
Let $f : \mathbb{R} \to \mathbb{R}$ be an increasing function, and let $a$ be a real number. Prove that $f$ is continuous at $a$ if and only if there exists a sequence $(a_n)_{n \ge 1}$ of positive real numbers such that
$$
\int_a^{a+a_n} f(x) \, dx + \int_a^{a-a_n} f(x) \, dx \le \frac{a_n}{n}, \quad n = 1, 2, 3, \dots
$$
$$
\int_a^{a+a_n} f(x) \, dx + \int_a^{a-a_n} f(x) \, dx \le \frac{a_n}{n}, \quad n = 1, 2, 3, \dots
$$