SOLUTION : (i)$f(x)=A x^2+B x+C \Rightarrow x \in I \text { and } f(x) \in I \\$
$\text { at } x=0, f(0)=C \Rightarrow C \text { is integer at } x=1, f(1)=A+B+C \\$
$\because \quad C \text { is integer } \therefore A+B \text { is also integer } \\$
$\text { at } x=-1, f(-1)=A-B+C \Rightarrow f(1)+f(-1)=2 A+2 C \\$
$\because \quad C \quad \text { is integer } \therefore 2 A \text { is also integer }$
(ii) $f(x)=A x(x-1)+(A+B) x+C \Rightarrow f(x)=2 A \frac{x(x-1)}{2}+(A+B) x+C$
If $x$ is an integer then $\frac{x(x-1)}{2}$ is also an integer and $2 A,(A+B), C \in I$ $\Rightarrow \quad f(x)$ is also an integer.