I have a function $f: (a,b) \to \mathbb R$. I need the function to be bounded on all closed subintervals of $(a,b)$ (of course the bound may be different for each subinterval).
This is the case for example if $f$ is continuous on $(a,b)$, or if it is monotonic.
Are there other conditions on $f$ that ensure that it is bounded on closed intervals? Ideally, what is the “most general” such condition?
The concept of local boundedness comes very close to what I wanted.
A function is locally bounded if for every point in its domain there is a neighbourhood where the function is bounded. This implies (linked reference, theorem 5) that the function is bounded on closed intervals. Also, conrtinuity implies local boundedness.