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I'm with this guy. Obviously I'm missing something or not understanding what is being said here, because to my head f(x) = x+1 changes every single point and is continuous.

Or in the glass of water point, what if I pour the water out on the table. It's certainly all been moved, yes?

(To be clear I'm obviously wrong / not understanding the premise, but I would be curious to know what I'm missing)

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It only applies to functions on bounded sets. The reals are not bounded. There are other restrictions, but that is one of them.

Your example function doesn't map any compact set back to itself.

Yep. I notice mention of this additional constraint from another comment. As far as I can tell the article is just incorrect since it misses this constraint up to that point, unless I'm just a clown and somehow not seeing it.

The upside here is that it seems perhaps I wasn't missing the premise, and it was just mis-stated :)

No you are right, quanta sucks. Im sure there are thousands of other people thinking of x+1 and scratching their heads

So add "mod p" to it.

Transformation is not continuous at the boundary p

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