Let \(X\) be a set1. A function \(f:X\to \mathbb R\), i.e. a function mapping this set to the set of real numbers is called
If \(f(X)\) is not bounded, (i.e. not bounded above or it is not bounded below or neither bounded above nor bounded below), we call \(f\) unbounded.
Corollaries: 1 2 3
Definitions: 4 5 6
Proofs: 9 10 11 12 13 14 15 16
Propositions: 17 18 19
Please note that $X$ does not necessarily have to be a subset of real numbers $\mathbb R$. The concept of bounded functions can be defined more generally for any kind sets. The only important detail is that the image set of the function is a subset of $\mathbb R$. ↩