When I was learning analysis (beyond, say, first-year calculus), one of the facts that most surprised me was the fact that there are functions that were smooth (i.e., infinitely differentiable) and yet compactly supported. Of course, I didn’t think about it with that phrasing; there’s a pretty simple geometric interpretation of smoothness for most functions one encounters in calculus (actually, one rarely sees differentiable functions that aren’t smooth!) Specifically, if a function isn’t smooth at , then there’s some sort of a “kink” at that point, or at least “around” that point.
Is this justified? Well, not totally, but let’s give a couple of examples to at least show why it’s a good first approximation. (more…)