- 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.