I am not sure if I understand exactly what a local or global maximum / minimum is.
That's why the following question:
Let $f$ be a function defined on the open interval $(-1, 1)$ with: $f : (-1, 1) \rightarrow \mathbb{R}$, $f(x) = x^3$
Does $f$ has a global or local maximum at $x = 1$? Can it be an extremum at all even though the function itself is not defined at $x$?