IMC 2021 · Problem 4

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function. Suppose that for every $\varepsilon > 0$, there exists a function $g \colon \mathbb{R} \to (0, \infty)$ such that for every pair $(x, y)$ of real numbers,

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

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