Discontinuous maps whose iterations are continuous

Let $X$ be a topological space and $f:X\to X$ a bijection. Let ${\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\mathcal C}(X,f)$ if and only if the bijection $f^n:X\to X$ is continuous. A subset $S$ of the set of integers ${\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\to X$ such that $S={\mathcal C}(X,f)$. A subset $S$ of ${\mathbb Z}$ containing 0 is called a submonoid of ${\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\mathbb Z}$ is realizable if and only if $S$ is a submonoid of ${\mathbb Z}$. Then we generalize this result to any submonoid in any group.

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