local extremum at endpoints

[Oneiromancy]

I thought local extremum did not exist at the endpoints of a closed bounded interval, however my textbook claims this.

Wikipedia:

"A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above). The neighborhood requirement precludes a local maximum or minimum at an endpoint of an interval."

[Dick]

If you consider a 'neighborhood' to be a 'neighborhood in the domain' your textbook is right. If you consider it to be a 'neighborhood in the reals' then Wikipedia is right. There are a lot of terms that are defined somewhat differently in different references. I think you'd better live by your textbooks definition. Until you change textbooks.