I'm studying Single Variable Calculus (E7) by James Stewart. In Chapter 4.1, the book does not have a clear statement about the relationship between absolute max/min and local max/min. This is my proposition:
Function $f$ is continuous on a closed interval $[a, b]$. If $c\in(a, b)$ and $f(c)$ is the absolute max/min, then $f(c)$ must also be a local max/min for any open interval containing c and within domain $[a, b]$.
Is this always true?