I have a floor function $f: \Bbb R → \Bbb Z$,
$$
f(x)=\lfloor x-7 \rfloor
$$
That I am trying to prove is surjective or onto. I know by definition that the floor function's domain is the set of reals and the range is the set of integers. I also know how to prove a function is surjective, but in this case I feel like I have hit a wall. I don't know how to proceed from here, any help?
Best Answer
To prove that a function is surjective, you need to prove that for each $y$ in the codomain there exists some $x$ (which will depend on the value of $y$ and will likely be represented as some function of $y$) such that $f(x)=y$.
Given some integer $y$, can we find some real number $x$ such that $\lfloor x - 7\rfloor = y$?