I am way out of my comfort zone here, so please be gentle!
I have been reading papers that try to use Artificial Neural Networks to approximate the Euler's Totient function of form $\varphi(n) = (p-1)(q-1), n = pq$. The factors here are primes. Now, obviously some of them mention the Universal Approximation Theorem (UAT) and the implications of showing that no way an ANN could learn $\varphi(n)$.
My question is, how do we even justify the hope of any success in an ANN learning $\varphi(n)$? All the versions of UAT I have seen require the function to be some kind of continuous. This is my concern, the totient function does not look continuous in any shape or form.
Then, I thought maybe it could be turned into a continuous function? I found this question. The maths presented there are definitely not something I'd understand in less than a day. So, could someone show me in relatively simple terms a somewhat continuous extension of $\varphi$ which we could hope to be learned by an ANN.
Or, the flip side that since an ANN can't even learn the continuous form, we can focus our efforts to showing that it may not be deterministic.