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>Isn't this mind-numbingly obvious?

Yeah, I think so.

>Collapse the shape into a function describing the area of the left side if you were to cut the shape at that X coordinate in that orientation. On one end the function has value zero and on the other its value is the area of the shape. Is there any reason why this function might not be continuous?

Not for this reason, though...

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Well, I'm listening.

Sorry, I wasn't challenging your logic, I was implying that it's intuitively obvious - it's funny to see the term "blindingly obvious" followed by a paragraph of formal logic.

Fair enough. I expressed it that because I was asking a genuine question, not being rhetorical. That was the argument I used to convince myself that my (unstated) intuition was indeed true and wanted to know if there was something I was missing.

For sure, proving things even just as a sanity check is definitely interesting (and sometimes things we think are "intuitive" turn out to be false or at least intractable to prove).

^blindingly^mind-numbingly^

Continuous is not necessary - a break in the area can still have a halfway point.

Um, sure, but my argument is that the function will always be continuous, as long as you're slicing a single shape.

not every orientation will be

For example?

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