Thanks for commenting—and for your patience. I’ve corrected the k/N slip-up, deleted the misleading clause about “a function not having any ‘jumps’ impl[ying] that it can’t have an ‘infinitely steep’ slope”, and thanked you at the bottom.
Now I have. (Sorry, apparently I got confused while trying to make parallel updates to the original and this mirrorpost; they’re not using the same source file due to differences between platform markup engines.)
a small change in the input can’t correspond to an arbitrarily large change in the output
The sign function doesn’t have an arbitrarily large change in the output. Do you maybe mean that an infinitesimal change in the input can only produce an infinitesimal change in the output? I don’t see how that fails, but maybe just because I don’t have a definition for it at hand.
Thanks for commenting—and for your patience. I’ve corrected the k/N slip-up, deleted the misleading clause about “a function not having any ‘jumps’ impl[ying] that it can’t have an ‘infinitely steep’ slope”, and thanked you at the bottom.
you haven’t deleted it.
Now I have. (Sorry, apparently I got confused while trying to make parallel updates to the original and this mirrorpost; they’re not using the same source file due to differences between platform markup engines.)
The sign function doesn’t have an arbitrarily large change in the output. Do you maybe mean that an infinitesimal change in the input can only produce an infinitesimal change in the output? I don’t see how that fails, but maybe just because I don’t have a definition for it at hand.