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The statement at compact support was that $f^{-1}(0)$ should be compact. I’ve corrected this.
Er, sorry. Not sure what happened there. Thanks for catching this.
I added some generalising remarks.
If I may ask, wouldn’t it be better to define “support of a function” (as done implicitly in compact support) and “compact” separately?
Probably. The general notion is more complicated than one might expect; see Wikipedia. But we can put it in bit by bit as needed.
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