One last post before I move on from ‘continuous’ curves….
I was going to include this in the last post, but that was already too long.
We have looked at curves which are continuous everywhere, and some which are not – but are continuous for most of the way.
Is it possible for a curve to be discontinuous everywhere? In theory yes, though we need to consider rational and irrational numbers.
A rational number is any that can be written down, accurately, with numbers. This includes numbers there are ‘recurring’ like 0.3333333 because this can be written as 1/3, and be accurate
Pi is an example of an irrational number
So if we say that y = f(x) where f(x) = 1 when x is rational and f(x) = 1/x where x is irrational… then that would define a curve, but one that is so chopped up is would be continuous in only very small sections between rational numbers