Is this book wrong about point B? I think it is an end-point but not an end-point minimum. The function at 0 has a lower value.
Book is correct here:
And end-point minimum if there exists a region in the domain for which B is an end-point (you agreed it's true) and for which $\displaystyle f(B)\leq x, \forall x $. Note: if exists a region, so it doesn't have to be for every region (that would be definition of minimum), but only for a region (in this example use $\displaystyle [3,4]$)
Could you please copy here how end-point is exactly defined?
And you are watching endpoints of $\displaystyle f(x)$, not every part of it. By that logic you could split function above (one that is defined on -2 to 4) split into infinite many parts and have infinite many end-points.
This is a good guide to piecewise functions and extrema: Powered by Google Docs