1. ## surjective function

How would I prove that f(x) = [x] is surjective, where [x] denotes the bracket function
I know how to prove that something is surjective, but I'm stuck on this particular example

2. ## Re: surjective function

What is the bracket function?

3. ## Re: surjective function

Originally Posted by emakarov
What is the bracket function?
Also known as the floor function, gauss brackets?

4. ## Re: surjective function

In proofs of surjection, knowing the domain and codomain is essential. For example, f(x) = [x] is not surjective if f is considered from $\mathbb{R}$ to $\mathbb{R}$, nor when it is considered from $\{x\in \mathbb{R}\mid x>0\}$ to $\mathbb{Z}$. So I'll assume that $f:\mathbb{R}\to\mathbb{Z}$.

We need to prove that for every $n\in\mathbb{Z}$ there is an $x\in\mathbb{R}$ such that [x] = n. Can you find several such x for n = 3?