Is this function surjective? $F:(0,1) \times \mathbb{Z} \ni (a,b) \rightarrow a+b \in \mathbb{R}$ 
Consider the function
$$F:(0,1) \times \mathbb{Z} \ni (a,b) \rightarrow a+b \in \mathbb{R}$$
Is it surjective?

At first I thought that $F$ is not surjective, because we will not get integer numbers. But, what if $0,(9)=1$?

Your first thought is correct.
If $a\in(0,1)$ and $b\in\mathbb Z$ then $a+b\in(b,b+1)$ and for every $b\in\mathbb Z$ we have $(b,b+1)\cap\mathbb Z=\varnothing$.
This proves that: $$\mathsf{im}F\cap\mathbb Z=\varnothing$$so the function is not surjective.

Be aware that:
$$0.99999\cdots=1\notin(0,1)$$
