Given $a, b \in\mathbb R$ with $−∞ < a \leqslant b < ∞$, every monotonic function in $\mathbb{R}^{[a,b]}$ $(f: [a,b] \longrightarrow \mathbb R$) is bounded.
Do we need the "monotonic" part there? Isn't every function that belongs to an interval whose domain is defined in that way bounded?
Thanks in advance
No. Take, for instance$$\begin{array}{ccc}[0,1]&\longrightarrow&\mathbb{R}\\x&\mapsto&\begin{cases}0&\text{ if }x=0\\\frac1x&\text{ otherwise.}\end{cases}\end{array}$$