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.