Surjective and Unbounded functions 
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*Every surjective function from $\mathbb{R}$ to $\mathbb{R}$ is unbounded.

*Every unbounded function from $\mathbb{R}$ to $\mathbb{R}$ is surjective.
Is it possible for either of these statements to be false? I have a feeling there is some counterexample that I am missing but I cannot figure it out.
My understanding is that if a function is unbounded then for all $M\in\mathbb{R}$ there is an $x$ such that $|f(x)| \gt M$.
And the definition of surjective is that for all $b \in Y$, there exists an $x \in X$ such that $f(x) = b$.
Clearly if we have some $M$ in the image of this function there is an $x$ that exists such that $f(x) = M$ by the definition of surjective.
I dont know if I am thinking of this correctly, intuition needed. Thanks.
 A: Let's be a bit clearer: a function $f\colon X\to\mathbb{R}$ is unbounded if and only if for every $M\in\mathbb{R}$ there exists $x\in X$ such that $|f(x)|\gt M$. The quantifier ("for every") is important; otherwise, you are just saying that there is at least one $M$ with the property.
It is certainly true that a surjective function onto $\mathbb{R}$ (regardless of the domain) is unbounded. Simply note that for every $M\in\mathbb{R}$, there exists $N\in\mathbb{R}$ with $N\gt M$, $N\gt 0$. Since surjectivity of $f$ implies that for every $y\in\mathbb{R}$ there exists $x$ such that $f(x)=y$, then putting $y=N$ shows that there exists $x$ such that $f(x)=N$, hence $|f(x)|=f(x)\gt M$.
So you are correct in the first one.
The second one, however, is false: unbounded means you can always exceed any given bound. But it does not guarantee that you can always "hit" every value.
Consider the greatest integer function, $f(x)=\lfloor x\rfloor$: this is defined as follows $f(x) = \max\{n\in\mathbb{Z}\mid n\leq x\}$. For example, $f(3.5) = 3$, $f(e) = 2$, $f(-1.5) = -2$, $f(\sqrt{2}) = 1$.
Is $f$ surjective? Is $f$ unbounded?
A: For the first question, let's prove the contrapositive: bounded implies not surjective. So let $f$ be bounded. This means that there is a number $M$ such $|f(x)|\le M$ for all $x$. But then the value $M+1$ is never taken by $f$ and thus $f$ is not surjective.
For the second question, the function $f(x)=x^2$ is unbounded but not surjective.
A: This statement 


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*Every unbounded function from R to R is surjective is false. Counter example as lhf points out, any function $f(x)=x^{4n}$ for $n \in \mathbb{N}$.

