I came across the following in some notes I'm looking at online...

"Iff(x,y) is continuous in some closed, bounded setDinR<sup>2</sup> then there are points inD, (x<sub>1</sub>,y<sub>1</sub>) and (x<sub>2</sub>,y<sub>2</sub>) so that f(x<sub>1</sub>,y<sub>1</sub>) is the absolute maximum and f(x<sub>2</sub>,y<sub>2</sub>) is the absolute minimum of the function inD."

I'm thinking that this doesn't always hold true. Am I correct? I think it wouldn't hold true for a flat plane (for example, z=1 or something), and it seems there could be a region with an absolute minimumormaximum, but not the other. For example, something with a 'dip' or a 'hill,' that slopes smoothly, and where the value at the edge of the region is the same on the entire edge. The examples seem pretty obvious, I'd just like to hear it from someone other than myself that there are exceptions to the rule...