First of all - what do you need the generalization for? What kind of quantity you would like to measure, what properties you would like to have, etc...

@JDH: Thanks. The $lim$ was used in informal context (see: edit note). No, I do not insist on the existence of $\lim f(x)$.

Uncountability says nothing about explicit construction (it need not to be recursive). For example $x^\alpha$ (for $\alpha\in\mathbb{R}$) form a uncountable set. Thanks for link to 'fast-growing hierarchy'.

@JDH: Actually, the use of set theory here is trivial.

@JDH & TZ & TI: I mistaken the order, but obviously it do not change anything. Thanks. @AKG: "Canonical" - "standard" - "natural" (no, it do not have a precise definition). @TI: Since its about big O notation (the problem itself isn't in ct, but asks about its common tool). @GE: Thanks!