KVPY SX SB Exam 2016 Question Paper

The graph of the function$\mathit{f}\left(\mathit{x}\right)=\mathit{x}+\frac{1}{8}\sin \left(2\mathit{\pi }\mathit{x}\right)$, $0\le \mathit{x}\le 1$ is shown below. Define ${\mathit{f}}_1\left(\mathit{x}\right)=\mathit{f}\left(\mathit{x}\right),{\mathit{f}}_{\mathit{n}+1}\left(\mathit{x}\right)=\mathit{f}\left({\mathit{f}}_{\mathit{n}}\left(\mathit{x}\right)\right)$, for n ≥ 1 Which of the following statements are true ?
I. There are infinitely many x ∈ [0, 1] for which $\underset{\mathit{n}\to \mathit{\infty }}{\lim }{\mathit{f}}_{\mathit{n}}\left(\mathit{x}\right)=0$
II. There are infinitely many x ∈ [0, 1] for which$\underset{\mathit{n}\to \mathit{\infty }}{\lim }{\mathit{f}}_{\mathit{n}}\left(\mathit{x}\right)=\frac{1}{2}$
III. There are infinitely many x ∈ [0, 1] for which $\underset{\mathit{n}\to \mathit{\infty }}{\lim }{\mathit{f}}_{\mathit{n}}\left(\mathit{x}\right)=1$
IV. There are infinitely many x ∈ [0, 1] for which $\underset{\mathit{n}\to \mathit{\infty }}{\lim }{\mathit{f}}_{\mathit{n}}\left(\mathit{x}\right)$ does not exist