How to argue some of following maps form $\mathbb Z\to Q$ are not possible?

Question

{$f:Z\to Q|$f is bijective and monotonically increasing}

Actually I only now that there is function bijective form $N\to N\times N$ I can make it from $Z\to N\times N$.

But after that I could not know.

I wanted to know that how to argue that map are not possible .

Any Help will be appreciated

Answer 1

Hint:

Use proof by contradiction

Hint 2:

Think about what value $z\in\mathbb Z$ maps to the number $\frac{f(0)+f(1)}{2}$. Remember, $a<\frac{a+b}{2}<b$ if $a<b$.