I’m not sure I like using the word ‘discontinuous’ to describe any real person’s position on plausible investment-output curves any longer; people seem to think it means “intermediate value theorem doesn’t apply,” (which seems reasonable) when usually hard/fast takeoff proponents really mean “intermediate value theorem still applies but the curve can be almost arbitrarily steep on certain subintervals.”
FWIW when I use the word discontinuous in these contexts, I’m almost always referring to the definition Katja Grace uses,
We say a technological discontinuity has occurred when a particular technological advance pushes some progress metric substantially above what would be expected based on extrapolating past progress. We measure the size of a discontinuity in terms of how many years of past progress would have been needed to produce the same improvement. We use judgment to decide how to extrapolate past progress.
This is quite different than the mathematical definition of continuous.
FWIW when I use the word discontinuous in these contexts, I’m almost always referring to the definition Katja Grace uses,
This is quite different than the mathematical definition of continuous.