International Mathematics Competition
for University Students
2017

IMC2017: Day 2, Problem 6

6. Let $f:[0;+\infty)\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to +\infty} f(x)=L$ exists (it may be finite or infinite). Prove that $$ \lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L. $$

Proposed by: Alexandr Bolbot, Novosibirsk State University

Hint: Replace the integral by $\int_0^nf$ and split it into two parts.

  Solution