1) Can every discrete function be approximated (better: interpolated) by some continuous function? By discrete function, I mean: there are countably infinitely many elements in the domain, and for each number in the domain, the function assigns a single number in the range. By approximation (interpolation), I mean that all input-output results must be maintained in a new function. The new function must be an extension of the old function, that is, the old one is a restriction of the new to the original domain. We can think of the new function as the original function on a domain enlarged with uncountably infinitely many elements.
2) can a "almost-continuous" function with finite discontinuities be nicely approximated by some continous function? (for the domain defined by "almost-continuous" function) By "nicely approximated", I mean that as domain variable $x$ increases to infinity, a new function's range variance from original function's range never exceeds some given some number.
Edit: for 2), by discontinuity, I refine it to be jump(step) discontinuity. And yes, by finite discontinuities, I mean finite number of discontinuities.
And what happens for removable discontinuity?
I ask for real-number cases and complex-number cases.
Edit: Just discard "discrete" from 1) and replace discrete function with the one that starts from "by approximation, ..."