@FedericoPoloni : $\lambda \to \infty$

and I shouldn't have even mentioned the word precision.

Suppose you were given $N$ samples of $f$ (samples are irregularly placed and being drawn from a countable dense set $D$), If you can construct $F_N$ such that $\limsup\limits_{N\to\infty} \|f-F_N\|_{L^{\infty}(\Omega)} = \epsilon$ you have learnt the function. This is what I should have got into but I ended up with this!

both the number of samples and there by the arithmetic precision, both should go to infinity and only then we can expect $\lim\limits_{N\to \infty}\|f-F_N\|_{L^{\infty}(\Omega)} \to \epsilon$ or ofcourse it can better. but this is the best thing that can be said. I should probably use limit supremum.

Agree, the finite arithmetic precision to represent data points $x_i$ is problematic. Even if you have a large number of data points, it won't make up for it.

Agree with you. This finite arithmetic precision to represent data points $x_i$ is problematic.

@Benoit Yes. We can just say uniform distribution over the domain...no need of the intermediate countable dense set D.

Just for illustration the solution function we got is $F(x) = x^2 + 5$. Then the computations required to compute the function $F$ is 1 multiplication and one addition. Just two computations and a couple of registers.

See this to know what I mean by compute a function : annals.math.princeton.edu/wp-content/uploads/…

"Compute a function" is an algorithm or a formula that gives out $F(x)$ for a given $x$. And to do this, for a single given $x$, need to be done with finite registers, computations and ...blah blah blah...