One of my Calculus lecture videos poses the following challenge (shortly after an exposition of the Extreme Value Theorem):
Construct a function $f$ such that
- $f$ is continuous on $(0, 1]$
- $f$ does not have a maximum on $(0, 1]$.
- $f$ does not have a minimum on $(0, 1]$.
After some thinking, I came up with the following function $f$, but on further thought I don't think it meets the requirements, because it's not continuous on its domain.
Does anyone know how to do it?
Solution or hints are both welcome (I'm not being graded for this)