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ICMC 2019/20 Round 2, Problem 3

Source: Imperial College Mathematics Competition 2019/20 - Round 2

August 7, 2020

college contests

Problem Statement

Let

\mathbb{R}

denote the set of real numbers. A subset

S\subseteq\mathbb{R}

is called dense if any non-empty open interval of

\mathbb{R}

contains at least one element in

S

. For a function

f:\mathbb{R}\to\mathbb{R}

, let

\mathcal{O}_f(x)

denote the set

\left\{x,f(x),f(f(x)),\ldots\right\}

.
(a) Is there a function

g:\mathbb{R}\to\mathbb{R}

, continuous everywhere in

\mathbb{R}

such that

\mathcal{O}_g(x)

is dense for all

x\in\mathbb{R}

for all

x\in\mathbb{R}

?
(b) Is there a function

h:\mathbb{R}\to\mathbb{R}

, continuous at all but a single

x_0\in\mathbb{R}

, such that

\mathcal{O}_h(x)

is dense for all

x\in\mathbb{R}

?
Proposed by the ICMC Problem Committee

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