I think following could be a choice function for choosing from powersets of $\mathbb{R} - \{\emptyset\}$ which is said to be only possible by axiom of choice. I am doing something wrong, esp. following is against axioms of set theory(may be infinity) but how? Or the definition of this function is wrong?
$f: \{x+\sqrt{2}/10^x : x \in \mathbb{N} \}= \{\ \{\sqrt{2}/10\}, \{1+\sqrt{2}/100\}, \{2+\sqrt{2}/1000\}, ... \}$
Edit: Actually, I want to choose from integer interval subsets of reals like; { [0,1), [1,2), [2,3),... }
(Sorry for my informal notation but you get the point I think)
No. It couldn't be. It is consistent with $\sf ZF$ that there is no such function, and therefore you cannot possibly define one explicitly.
You are also defining the function on a very specific collection of sets, it seems. There are a lot more sets which do not have this form. What is the choice from them?
If you want a concrete challenge, fix a bijection $f$ between $\Bbb R$ and $\Bbb{R^N}$, the set of all infinite sequences of real numbers, now for each countable set $A$ let $F_A=\{x\in\Bbb R \mid A=\operatorname{rng}(f(x))\}$, namely $F_A$ is all those sequences which enumerate (with repeating) exactly all the elements of $A$.
I dare you to find an explicit choice function from $\{F_A\mid A\subseteq\Bbb R \text{ and countable}\}$.