We can also consider a similar function f_epsilon(x) that returns the sum of x and a uniformly random number between 1 and 2 chosen at compile time, but constant at run time.
I agree it can't be specified in finite space.
The dichotomy is the computational model, from a utilitarian perspective we consider every conceptual computer to be a huge but finite finite-state-machine. One could abstractly (and less down-to-earth usefully) define/conceive of a computer that can store arbitrary variables representing uncountable objects (like real numbers etc)
If it's countable, then your computer can now reflect the previously-computable numbers across these epsilons, and give you new numbers, but you're still only covering 0% of the reals.
If it's uncountable, covering a range, then you just moved the problem back a step. On top of that, such a computer doesn't even have to do real work. You can spend 20 seconds using grade school arithmetic to map the input range to the entire set of reals, 1:1. But mapping a range of reals into a bigger range of reals is a pretty lousy definition of "computable".
Let me present a challenge to you: Pick a uniformly random number between 1 and 2. Now tell me what it is.
No 'coincidences', like getting exactly 1.34 or the square root of three. It has to be properly random.
The odds are 100% that you picked a 'normal' number that goes on forever with no pattern. A number that cannot be specified in finite space. In other words, a number that can't be computed.