my math teacher opened up a discussion about continuity and whether a single point is a continuous function and other stuff, and then after a couple minutes she was like âokay thatâs about all I have to say. letâs get back to calculusâ but I and the other two dudes at my table had way more to say and we just kept whispering at each other like
âlook, you canât have a limit from either side, let alone both, if itâs only one pointâ
âyeah but you can think of the point as just an infinitely small line segment, here, thereâs a specific definition of continuity for finite ranges
âinfinities arenât a good argument for that when youâre limiting it down to nothing. thereâs no way a point is continuous because thereâs nothing to continueâ
âyou can draw it without picking up your penâ
âyou wouldnât say the cantor series is continuous, would you?? it doesnât even matter, because points and piecewise functions arenât even really functionsâ
âyeah they literally areâ
âthey might fit the definition, but they canât actually exist mathematically if we canât describe them with an equation, you know?â
âfunctions and real-life possibilities donât always have to match up! like if thereâs an infinitely small holeâ
âokay but do you guys think discontinuities are even real, or are they a product of our number system?â
âour number system, obviously. thereâs no need for zero to exist except as a placeholder in writing other numbersâ
âthatâs not true! zero is completely realâ
âthen why havenât people figured out how to divide by it? our inability to divide by zero is whatâs causing all the discontinuities in the first placeâ
and the teacher just kept being like âif you guys wonât stop, can you at least keep it down?â