Question 65
Let $n$ be a positive integer, and let $f: [0, 1] \to \mathbb{R}$ be an integrable function. Show that there exists a point $a_n$ in the closed interval $[0, 1 - 1/n]$ such that either
$$
\int_{a_n}^{a_n+1/n} f(x) \, dx = 0 \quad \text{or} \quad \int_{0}^{a_n} f(x) \, dx = \int_{a_n+1/n}^{1} f(x) \, dx.
$$
$$
\int_{a_n}^{a_n+1/n} f(x) \, dx = 0 \quad \text{or} \quad \int_{0}^{a_n} f(x) \, dx = \int_{a_n+1/n}^{1} f(x) \, dx.
$$