International Mathematics Competition
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2021

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IMC 2021
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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: Start from a segment in place of \(\displaystyle \mathbb R\) and use its compactness. Or recall the cool things called ``the Lebesgue characterization theorem'' and ``the Baire characterization theorem''.

    

IMC
2021

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