International Mathematics Competition
for University Students
2021

IMC2021: Day 1, Problem 4

Problem 4. Let $\displaystyle f:\RR\to\RR$ be a function. Suppose that for every $\displaystyle \varepsilon > 0$, there exists a function $\displaystyle g:\RR\to(0,\infty)$such that for every pair $\displaystyle (x,y)$ of real numbers,

$\displaystyle \text{if} \quad |x-y| < \min\big\{g(x),g(y)\big\}, \quad\text{then}\quad \big|f(x) - f(y)\big| < \varepsilon.$

Prove that $\displaystyle f$ is the pointwise limit of a sequence of continuous $\displaystyle \RR\to\RR$ functions, i.e., there is a sequence $\displaystyle h_1,h_2,\ldots$ of continuous $\displaystyle \RR\to\RR$ functions such that $\displaystyle \lim\limits_{n\to\infty}h_n(x)=f(x)$ for every $\displaystyle x\in\RR$.

Camille Mau, Nanyang Technological University, Singapore

  Hint    Solution