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IMC 2021 first day , problem 4

Source: IMC first day , problem 4

August 4, 2021
functionreal analysisIMC 2021

Problem Statement

Let f:R→Rf:\mathbb{R}\to \mathbb{R} be a function. Suppose that for every ε>0\varepsilon >0 , there exists a function g:R→(0,∞)g:\mathbb{R}\to (0,\infty) such that for every pair (x,y)(x,y) of real numbers, if ∣x−y∣<min{g(x),g(y)}|x-y|<\text{min}\{g(x),g(y)\}, then ∣f(x)−f(y)∣<ε|f(x)-f(y)|<\varepsilon Prove that ff is pointwise limit of a squence of continuous R→R\mathbb{R}\to \mathbb{R} functions i.e., there is a squence h1,h2,...,h_1,h_2,..., of continuous R→R\mathbb{R}\to \mathbb{R} such that lim⁡n→∞hn(x)=f(x)\lim_{n\to \infty}h_n(x)=f(x) for every x∈Rx\in \mathbb{R}
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