IMC 2021 Problem 4

Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose that for every $\varepsilon >0$ , there exists a function $g:\mathbb{R}\to (0,\infty)$ such that for every pair $(x,y)$ of real numbers, if $|x-y|<\text{min}\{g(x),g(y)\}$ , then $|f(x)-f(y)|<\varepsilon$ Prove that $f$ is pointwise limit of a squence of continuous $\mathbb{R}\to \mathbb{R}$ functions i.e., there is a squence $h_1,h_2,...,$ of continuous $\mathbb{R}\to \mathbb{R}$ such that $\lim_{n\to \infty}h_n(x)=f(x)$ for every $x\in \mathbb{R}$