is there any example of function which is computable on some set and uncomputable on other set? That is for example function f(n) which is computable on some (finite, or for example for even numbers) set A of N and, uncomputable on N\A ?


By nontrivial example I mean function which is not defined as computable function for set A and uncomputable for set N\A that is by use of the "if $x \in A$ then ..." statement, but by one, given procedure or definition. If You have a problem what is mean "one procedure or definition" take an assumption that in definition of such function do not appear sentence "if $x \in A$ then ..." 



Nontrivial, partially uncomputable function
How about the characteristic function of the Halting Problem? That is, $f(x)=1$ iff $\Phi\_x(x)\downarrow$ and $f(x)=0$ otherwise.


This function is clearly incomputable, but is computable on large infinite sets since by the padding lemma we can produce large (computable) sets $X$ of indices for equivalent programs which we already know halt or diverge.


