Question about describing a sub-group of $S_{\mathbb{R}}$

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We had the function $f(x)=x+1$.

What is $\langle f\rangle$ of $S_{\mathbb{R}}$?

How can I describe $\langle f\rangle$?

Thank you!

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$f^n(x)=x+n$, $f^{-1}(x)=x-1$, so $\langle f\rangle =\{x+n:n\in\Bbb Z\}$.