Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases:
$$\begin{aligned}
& \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\
& \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12
\end{aligned}$$
In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?\\
(A) In both cases.\\
(B) In neither case.\\
(C) In Case 1. but not necessarily in Case 2.\\
(D) In Case 2. but not necessarily in Case 1.