I left out a bunch of fascinating details
What do you do if the denominator c + dx might be zero?
What if z makes a mistake and outputs the wrong number?
What if you want z to output decimal digits instead of continued fraction terms?
What if you want z to guess what is coming up without seeing it?
(And why would you want z to do that?)
How can you represent a number like π, whose continued fraction is unknown?
How do you get around the need to represent arbitrarily large terms?
How do you extract a square root?
Sorry, I only got an hour!